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Your Host - Darrel Chaney
The United Way of White County Celebrity Golf Committee wants to say Thank You "Darrel Chaney" for your continued leadership with the United Way Celebrity Golf tournament. Darrel started this tournament over 15 years ago with a couple of players. To date, this tournament has become one of the most popular tournaments in North Georgia, raising over 60K each year for United Way of White County. Celebrities come from all over the country each year to participate. The committee has changed the name of this tournament in recognition of Darrel to "The Darrel Chaney Invitational".
"The Darrel Chaney Invitational"
Benefitting the United Way of White County
Darrel Chaney, born March 9, 1948, in Hammond, Indiana, is an American former player/announcer in Major League Baseball who played for the Cincinnati Reds and Atlanta Braves from 1969 to 1979. In the early 1980s he worked for the Braves as an announcer along with Ernie Johnson, Skip Caray and Pete Van Wieren. He was on the Atlanta Braves Radio Network as well as WTBS-TV.
Chaney was a graduate of Morton High School in Hammond, Indiana, where he was a three-sport athlete and an All-American football player and was named the Northwest Indiana Times Athlete of the Year in 1966. His wife Cindy is also from Hammond and is a graduate of George Rogers Clark High School. He had several football scholarship offers from Big Ten schools but signed with Ball State University because there he could play both football and baseball. However, he was selected by the Reds in the second round of the 1966 draft. Darrel broke through and led the Southern League with 23 home runs in 1968, earning him a spot on the Reds' roster in 1969. Chaney continued with the Reds in the 1970s. Darrel played in three World Series for the Reds' "Big Red Machine" teams, in 1970 and 1972 and on the World Series-winning team of 1975. Chaney said he was fortunate to be part of the "Big Red Machine," the team included Pete Rose, Johnny Bench, Tony Perez and Joe Morgan."We were the best team in that era and I was glad and fortunate to be a part of it," Chaney said. "Three World Series and one world championship. Not bad." Chaney was traded to Atlanta after the 1975 season and in 1976 batted .252 with one home run and 50 RBI as the Braves' regular shortstop. In 915 career games, Chaney hit for a .217 batting average, with 14 home runs, 190 runs batted in, 237 runs scored, 458 hits, 75 doubles, 17 triples and 19 stolen bases.
It is because of Darrel Chaney that we have the pleasure to enjoy this tournament every year and without him this event would not be possible. Your host Darrel works many hours to ensure this event is the best it can possibly be. Thank you Darrel.
Darrel has been a motivational and inspirational speaker since 1969 and is available to speak on a wide range of topics. From church groups to corporate meetings to national conventions and appearances,
Darrel will inspire and motivate your audience to recognize their own significance.
You can read more about Darrel and watch a short video presentation on his website. www.darrelchaney.com


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ranges from -14% to 16%
Princeton University Doctoral NEW JERSEY $248,000 4.1% 490 $320,000 92.9
Columbia University in the City of New York Doctoral NEW YORK $259,700 3.5% 766 $316,500 88.4
Stanford University Doctoral CALIFORNIA $256,100 4.7% 667 $312,600 94.7
Harvard University Doctoral MASSACHUSETTS $244,300 3.4% 1 019 $305,500 92.5
New York University Doctoral NEW YORK $218,300 2.5% 882 $290,700 89.7
University of Chicago Doctoral ILLINOIS $241,900 N/A 614 $288,900 85.7
Massachusetts Institute of Technology Doctoral MASSACHUSETTS $232,200 4.4% 637 $287,600 102.0
University of Pennsylvania Doctoral PENNSYLVANIA $223,600 3.7% 636 $286,600 92.4
University of California-Los Angeles Doctoral CALIFORNIA $214,000 N/A 951 $284,100 91.0
Yale University Doctoral CONNECTICUT $230,900 4.9% 614 $278,500 94.7

redpajama_set_name: RedPajamaCommonCrawl


text:
Paula Kahumbu is a wildlife conservationist and chief executive officer of WildlifeDirect. She is best known as a campaigner for elephants and wildlife, spearheading the Hands Off Our Elephants Campaign, which was launched in 2014 with Kenyan First Lady Margaret Kenyatta. She has recently in 2022 been appointed as the first National Geographical Explorer as a board of Trustees member at the National Geographic Society.

Education and early career
Kahumbu was born on June 25, 1966, and grew up in Nairobi, Kenya, and there attended  primary and secondary school at Loreto Convent Msongari. She was first mentored by the renowned conservationist Richard Leakey. She was granted a Government of Kenya Scholarship to study Ecology and Biology at University of Bristol. She then received her master's degree at University of Florida in Wildlife and Range Science in 1992. Her early studies and fieldwork centered on primates, writing her Masters thesis on the monkeys of the Tana River Primate National Reserve.

Between undergraduate and graduate school, Kahumbu returned to Kenya to work for the Kenya Wildlife Service. There, she was charged with counting and measuring the ivory stockpile in the country's vaults in preparation for Richard Leakey's now-famous internationally televised burning of the tusks. The event changed her focus from primates to elephants of her doctoral work. She was granted a Petri scholarship to attend Princeton University to complete her PhD in Ecology and Evolutionary Biology from 1994 – 2002, where she studied elephants in the Shimba Hills on the Kenyan coast. In 2005, she received her Certificate in the Program for Management Development through the Gordon Institute of Business Sciences at University of Pretoria.

Conservation career
After receiving her PhD, Kahumbu returned to the Kenya Wildlife Service and led the Kenyan delegation to the Convention on International Trade in Endangered Species. In 2007, Kahumbu became the executive director of WildlifeDirect, a nonprofit organization co-founded in 2004 by her mentor Richard Leakey as an online platform to provide voice to African conservationists. The organization has since become the largest wildlife blogging site in Africa and has covered a diverse array of conservation issues—from protecting chimpanzees in Sierra Leone to African Painted Dogs in Zimbabwe. Those who are inspired by the blog posts can then donate directly to the conservationists on the ground, circumventing administrative fees.

The "Hands Off Our Elephants" campaign launched by Kahumbu at WildlifeDirect to put an end to the poaching of elephants and trafficking of ivory, and has since gained the support of Kenyan First Lady Margaret Kenyatta. The campaign aims to leverage media to drive behavior change and to foster agency within the local community to take action. In tandem, the campaign has worked to propose concrete legislation to enforce against the illegal ivory trade. When the campaign began in 2014, more than 100,000 elephants across the African continent had been killed for their ivory over three years.

Kahumbu also teaches conservation to students as a lecturer at Princeton University, where she leads an undergraduate course in community conservation during an annual field course in Kenya.

Public engagement
Kahumbu has reached the public through a variety of mediums to advocate for conservation—from television to editorials to museums. She is the producer of NTV Wild, an award-winning wildlife documentary series, and NTV Wild Talk, a television show hosted by Smriti Vidyarthi that covers conservation issues. She has also contributed regularly to The Guardian, advocating for protection of elephants through a solutions-based approach—from ending corruption to inspiring a young generation of conservationists—and denouncing the ivory trade. Kahumbu is also a chairperson for National Museums of Kenya.

Kahumbu is also an accomplished children's book writer, co-authoring global best sellers like Owen and Mzee: The True Story of a Remarkable Friendship, based on the unlikely friendship of hippopotamus and an Aldabra giant tortoise Owen and Mzee.

Awards and honors

 Rolex National Geographic Explorer of the Year June 2021
National Geographic Buffet Award Winner for conservation leadership in Africa, 2011
 National Geographic Emerging Explorer Award, 2011
Special Commendation, United Nations Person of the Year Award, 2013
Order of the Grand Warrior Award, Kenyan Ministry of Environment, Water and Natural Resources, 2014
Whitley Award Winner, donated by The LJC Fund in memory of Anthea and Lindsey Turner, 2014
Round Square Idealist

References

Women conservationists
Princeton University alumni
National Geographic people
Living people
Women ecologists
Nonprofit chief executives
21st-century Kenyan women scientists
21st-century Kenyan scientists
Year of birth missing (living people)
People from Nairobi
21st-century Kenyan businesswomen
21st-century Kenyan businesspeople

redpajama_set_name: RedPajamaWikipedia


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text:
Карліс Янсонс (, 23 квітня 1896, Нітаурська волость, Латвія - 16 червень 1986 року, Цесіс,  Латвія) — латвійський скульптор.

Біографія
З чотирьох років жив в Цесісі. Освіту здобув в реальному училищі в Цесісі.

Під час  Першої світової війни продовжив навчання в комерційних училищах в  Казані і Пскові. У 1917 році вступив до Ризького політехнічного інституту, щоб вивчати архітектуру. Спочатку практикувався у  живописі, але в 1918 році зосередився на скульптурі.

До 1925 року навчався в Латвійській академії мистецтв, учень професора Костянтина Рончевського (Rončevska).

З 1927 року брав участь у виставках, що влаштовуються в Ризі і в інших містах, за кордоном, мав персональні виставки в Цесісі.

До 1972 року працював в Академії мистецтв, викладав скульптуру.

Похований на Лісовому цвинтарі в Цесісі.

Син - скульптор Андрейс Янсонс (1937-2006).

Відомі роботи

Створив пам'ятник визволителям Єлгави (1932), пам'ятник Резекненської Мари (1939) і партизанському полку в Балві (1938). За радянських часів ці роботи були знищені і відновлені сином Янсонса Андрейсом після відновлення незалежності Латвії.

Він також створив багато пам'ятників, у тому числі — першому президенту Латвії Янісу Чаксте на Лісовому цвинтарі в Ризі.

Джерела
 Karlis Jansonas. Visuotinė lietuvių enciklopedija, T. XIII (Leo-Magazyn). – Vilnius: Mokslo ir enciklopedijų leidybos institutas, 2008. 525 psl.

Латвійські скульптори
Уродженці Латвії


For better or worse, I can sit down at the computer and bang away on the keyboard for hours without taking a break. Doing this regularly isn't healthy, so I wanted an unobtrusive app to automatically remind me to take breaks.
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You can specify how often and long you want your breaks.


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text:
/* This file provided by Facebook is for non-commercial testing and evaluation
 * purposes only.  Facebook reserves all rights not expressly granted.
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 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
 * FACEBOOK BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
 * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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- (ASCellNode *)collectionView:(ASCollectionView *)collectionView nodeForItemAtIndexPath:(NSIndexPath *)indexPath
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  NSString *text = [NSString stringWithFormat:@"[%ld.%ld] says hi", indexPath.section, indexPath.item];
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redpajama_set_name: RedPajamaGithub


If you're already in our Discord server, you'll know that this week's AMA will be with Videomantic!
He's our head editor, and edits CSG, DD and BTTH. He handles editor recruitment, and is also one of Gravity's oldest members, having joined on the 26th of March 2015. Feel free to ask him anything about editing, or how Gravity has changed during his time as part of it. You might want to know how long it takes him to edit a chapter, how many editors we have, what the challenges of his role are, etc.
If you don't know what timezone it'll be for you, go here and type in your location, then compare it to 7:10pm EDT.
If you aren't in our Discord server, click here to join!


Once Indra, the King of Gods, while riding on an elephant came across a sage named Durvasa who offered him a special garland. Indra accepted the garland but put in on the trunk of the elephant. The elephant was irritated by the smell and it threw the garland on the floor. This enraged the sage as the garland was a dwelling of Sri (fortune) and was to be treated as prasada. Durvasa Muni cursed Indra and all devas to be bereft of all strength, energy, and fortune.
In battles that followed this incident, Devas were defeated and Asuras (demons) led by king Bali, gained control of the universe. Devas sought help from Lord Vishnu who advised them to treat asuras in a diplomatic manner. Devas formed an alliance with asuras to jointly churn the ocean for the nectar of immortality and to share it among them. However, Lord Vishu told Devas that he would arrange that they alone obtain the nectar.
The churning of the Ocean of Milk was an elaborate process. Mount Mandaranchal was used as the dasher (churning tool), and Vasuki, the king of serpents, became the churning rope. The gods held the tail of the snake, while the demons (Asuras) held its head, and they pulled on it alternately causing the mountain to rotate, which in turn churned the ocean. However, once the mountain was placed on the ocean, it began to sink. Vishnu in his second incarnation, in the form of a turtle Kurma, came to their rescue and supported the mountain on his back.
As per (Hari Bhakti Vilasa 5/257 from Skanda Purana , 258/259 From Srimad Bhagvatam 11. 27. 12-14) a Shaligram Shila with Golden flow on a Shaligram Shila will represent the Nector that came out after the Samundra Manthan. These Shilas are very few on this Universe as where ever the drop of Nector fell, the Shaligram Shila has absorbed that nector and same has become golden flow and emerged out from the Shila. This Shila is Samundra Manthan Shila. There will be several 'Akrutis' of various Devetas and Demaons and the 14 Ratnas that emerged out during this churing of ocean that can be seen on the Shaligram Shila. The Shila blesses the devotee with spiritual enlightenment and rejunivation of positive energies and discarding of negative energies. This enlightens the consciousness like the wave of Ocean. It gives tremendous concentration and stability in thoughts like Mandara mountain. Vasuki represents the desires and we need to control our desires. The poison represents the difficulties, sufferings and pain in life and worshipper comes out of these difficulties by Karma Yog, Gyan Yog, Dhyan yog and Bhakti yog. Lord Shiva symbolises the will to take everybody's responsibility on one's shoulders, the awakening of "Super Conciousness". Dhanvantri symbolises the good health and longitivity. The Nector symbolises the achieving of ultimate goal, the Salvation and Laxmi represents the universal enrichment and wealth to do good and selfless service to others.


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text:
The Best Research on Body Fat, Losing Fat, and Eating Fat
Tom Kelso
St. Louis, Illinois, United States
If I see an article on the Internet or in paper publication worth reading, I take notes of the gist of the article and secure its reference. If it contains valuable and practical information, then it is worthy of passing on to others.
I've accumulated a few such articles over the last months. What follows is a brief synopsis of these articles and the related research links. I'm hopeful they give you a greater insight to facilitate better training and nutrition.
For the Best Fat Loss, Don't Run
Nate Miyaki wrote an excellent piece on T-Nation regarding cardio training for fat loss. Let's take a look at what we can learn from him and others who share his opinion.
When it comes to running, people typically have one or more of these reasons:
Improved sport performance
General performance enhancement and health benefits
Stress relief obtained by the endorphin rush
A love of running and competing
These are all fine justifications for running. If it makes you happy, go for it. But if you're running as a means of reducing body fat, you have better options.
"The key is to elevate your resting metabolic rate (via anaerobic work) to promote better post-workout fat burning and muscle building."
Look at true runners - marathoners. Their training is primarily (relatively speaking) low-intensity and non-muscle-building. No one is tackling or colliding with them. They don't normally find themselves in compromised positions (e.g., changing multiple directions, landing awkwardly). They're in control of their mostly straight-line task.
But distance running is a contact sport. It involves thousands of single-leg collisions with the ground. Over time, it creates havoc on your joints.
In his article, Miyaki expounded on the downside of cardio and why traditional cardio is poor for fat loss. The majority of your training should be anaerobic (strength training/interval training), as opposed to aerobic (traditional lengthy and low-level) in nature.
Attempting to alter your physique goes beyond simple calories in versus calories out. The key is to elevate your resting metabolic rate (via anaerobic work) to promote better post-workout fat burning and muscle building. And yes, diet is important here.
Strength training, high-intensity interval training (HIIT), and anaerobic-based circuit training are much better options than long-distance running or plodding away on a treadmill for an hour.
The calories in versus calories out faction run into issues. Over-the-top aerobic exercise coupled with a large reduction of calories consumed leads to decreased muscle tissue (the muscle gets used as energy) and the tendency to store more body fat as a survival mechanism. This is not good.
For optimal fat loss and lean tissue gains, go in this order of priorities:
Cardio - interval training
Cardio - long, steady state
The Research on High-Intensity Interval Exercise
Unless you have had your head buried in the ground like an ostrich, you should by now have figured out high-effort interval training is a spectacular exercise option. (I'll admit, I've been to the zoo and have yet to see an ostrich with its head in the ground, so maybe it's time to use another analogy. Either way, you get my point.)
"If the benefits of a forty-minute training session can be obtained from a twenty-minute higher-effort training session, why would you not opt for the more time-efficient session?"
Brief repeated sessions of high-intensity interval exercise garner changes that are elicited by traditional endurance training. Think about it. If the benefits of a forty-minute training session can be obtained from a twenty-minute higher-effort training session, why would you not opt for the more time-efficient session?
Most studies use prolonged exercise involvement; however, but this 2009 study published in Applied Physiology, Nutrition, and Metabolism showed a small dose of intense interval exercise - equivalent to two-minutes of all-out cycling - was sufficient to improve gains. As little as six sessions of HIIT performed over two weeks - a total of fifteen minutes of all-out effort - increased maximal activity of the cell factory (mitochondria) and improved performance during tasks that previously relied on aerobic energy activation. HIIT increased the potential for glucose and fatty acid oxidation. Wow.
"[A] small dose of intense interval exercise - equivalent to two-minutes of all-out cycling - was sufficient to improve gains."
The Battle Over Fat Consumption
Authority Nutrition put together a great set of facts explaining how the war against saturated fat has to be one of the largest misconceptions in the annals of human nutrition. It's an emotional issue with both sides strongly hanging on to their arguments based on the science that is promulgated to everyone, whether it is peer-reviewed or not.
So, good gracious, what should we take as proven facts and proven crap? How about these discussion points?
Over the years that humans have reduced the intake of animal fat and cholesterol, numerous diseases have increased. Currently, we are experiencing a worldwide endemic of obesity, metabolic syndrome, and type 2 diabetes.
Over the past few decades, studies conclusively showed the culprit of heart disease is neither saturated fat nor dietary cholesterol.
The low-fat recommendations were based on specious studies that have been thoroughly proven wrong. (So that package of low-fat Sun Chips may be okay after all. Maybe.)
Europeans consuming the most saturated fats have the lowest risk of heart disease.
In 1977, the low-fat diet was promoted to all Americans. The obesity epidemic started at almost the exact same time these low-fat guidelines were revealed.
Courtesy of National Center for Health Statistics (US). "Health, United States, 2008: With Special Feature on the Health of Young Adults." Hyattsville (MD): National Center for Health Statistics (US); 2009 Mar. Chartbook.
Many large studies have concluded low-fat diets do not cause weight loss and have a negligible effect on long-term cardiovascular disease. Women who ate a high-fat and low-carbohydrate diet until satiated actually lost twice as much weight as compared to those who ate a restricted low-fat diet. Additionally, all major risk factors for heart disease and diabetes declined.
Courtesy of Brehm BJ, et al. "A randomized trial comparing a very low carbohydrate diet and a calorie-restricted low fat diet on body weight and cardiovascular risk factors in healthy women." The Journal of Clinical Endocrinology & Metabolism, 2003.
The heart disease epidemic began in 1930. The obesity epidemic started in 1977. Around 1990 is when the diabetes epidemic began. These diseases have increased even more as animal fats have been replaced with vegetable oils and trans fats.
When saturated fat began getting a bad rap relative to heart disease, high-fat dairy products such as butter were considered evil. Ironically, like with other nutrition beliefs, it turns out saturated fat is undamaging and trans fat is more dangerous.
Courtesy of Hu FB, et al. "Trends in the Incidence of Coronary Heart Disease and Changes in Diet and Lifestyle in Women." The New England Journal of Medicine, 2000.
My Advice to You
So there you have it - some solid science-based advice and research for you to sit down and spend some time with. My advice is to read as much as you can, study the facts, and be vigilant. What you have been doing may not be working, and what you thought wouldn't work may have merit to it.
Check out these related articles on training, fat loss, and health:
The Top 10 Best Training Exercises
Why Steady State Cardio for Fat Loss Is a Bad Decision
Go Anaerobic: What It Is and Why to Do It
Folate, Alcohol, and Saturated Fat Benefit HDL Levels
1. Miyaki, N. "The Best Damn Cardio Article - Period. " T Nation. 14th October, 2014.
2. Gibala M., "Molecular responses to high-intensity interval exercise. "Appl Physiol Nutr Metab. 2009 Jun;34(3):428-32. doi: 10.1139/H09-046.
3. Gunnars, K., "6 Graphs That Show Why The 'War' on Fat Was a Huge Mistake." Accessed January 7th, 2014.
4. Siri-Tarino, PW. et al., "Meta-analysis of prospective cohort studies evaluating the association of saturated fat with cardiovascular disease. "Am J Clin Nutr January 2010 ajcn.27725
5. Rong, Y. et al., "Egg consumption and risk of coronary heart disease and stroke: dose-response meta-analysis of prospective cohort studies." BMJ 2013;346:e8539
6. Howard BV. et al., "Low-fat dietary pattern and weight change over 7 years: the Women's Health Initiative Dietary Modification Trial. " JAMA. 2006 Jan 4;295(1):39-49.
7. No authors listed. "Multiple Risk Factor Intervention Trial: Risk Factor Changes and Mortality Results." JAMA. 1982;248(12):1465-1477.
8. Brehm BJ. et al., "A Randomized Trial Comparing a Very Low Carbohydrate Diet and a Calorie-Restricted Low Fat Diet on Body Weight and Cardiovascular Risk Factors in Healthy Women" J Clin Endocrinol Metab. 2003 Apr;88(4):1617-23.
9. Lopez-Garcia, E., et al, "Consumption of Trans Fatty Acids Is Related to Plasma Biomarkers of Inflammation and Endothelial Dysfunction. " J. Nutr. March 1, 2005 vol. 135 no. 3 562-566
Photos courtesy of CrossFit Empirical.
See more about: cardio, body fat, fat loss, HIIT, saturated fats, calories, trans fats
The Complete Pregnancy
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Strength Training For Beginners
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Push, Pull and Thrive
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Coach of the Decade: Greg Glassman, CrossFit's Founder
The Pros and Cons of Intuitive Eating and Body Composition
10 Tips to Restarting Fitness After Childbirth
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Choose the Right Weight Class for Olympic Weightlifting
How Much: Time, Quality, and Quantity


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Throughout my life, I have struggled with anxiety. In my younger years, most of it was a result of social discomfort and a fear of other people. I didn't speak unless I was spoken to and I rejected the world before it got the chance to reject me.
I was the little girl in the corner using a chapter book as a shield and finding new ways to disappear. As of the past few years, my worries have shifted. As I've gotten older, much of my anxiety has been directly related to my identities and what they mean to these other people. I'm afraid of what "me" means to "them". It overwhelms me to think of all the reasons why people deny each other humanity and the weight of the target on your back can be haunting.
So it seems that now I am afraid of myself.
I've always had a hard time articulating this sense of inherent danger in one's body to those who may not sit at the same intersections as I do, and when I try I often leave having been made to feel as if I'm being paranoid. But in the wake of the Orlando shooting, I have officially been hit with the true weight of my fears: their validity. I was right to be afraid. And I wish I wasn't. This is no insightful revelation. Anyone paying attention would've learned fear like I did. The lesson is always the same.
1) Bodies like mine and people like me are magnets for violence. Brutality never forgets our address.
2) Safe spaces are conditional. Refuge can be confiscated. Churches aren't safe. Clinics aren't safe. Clubs aren't safe.


Black Urad dal whole 150 gms.
1. Pick and wash whole black urad. Soak it in 4 cups of water for 8-10 hours.
2. Cook it in 4-5 cups of water along with salt, red chili powder, Ginger Paste and Garlic Paste. Bring it to a boil. Reduce flame and simmer for about an hour or till it is completely cooked and tender.
3. Add tomato puree, butter and Garam Masala Powder and cook on a slow flame for an hour.
4. Add fresh cream, correct seasoning and simmer for another 10 minutes.
5. Serve hot with a bread of your choice.


text:
Резе́рвне (до 1945 року — Кючю́к-Мускомія; ) — село в Україні, у Балаклавському районі Севастопольської міськради. Населення становить 134 особи.

Населені пункти Балаклавського району


Published: August 6, 2014 12:00 am Updated: Aug. 6, 2014 3:13 p.m.
Nathaniel Kibby listens during a court hearing Wednesday Aug. 6, 2014, is Ossipee, N.H. Kibby is charged with kidnapping 14-year-old Abigail Hernandez as she walked home from high school. Abigail, now 15, returned home last month.
OSSIPEE, N.H. — A judge on Wednesday put off ruling on whether to move evidence including the house of a man charged with kidnapping a 14-year-old girl who returned home after nine months.
"We're entitled to do our own testing, come up with our own photographs and diagrams," Friedman said.
Young said that posting a guard on the property round-the-clock would be too expensive and noted that the case's compelling elements and the national media attention will attract gawkers and intruders.
"The state can post someone up at the property," Friedman countered. "They may not want to. It may be expensive. That's not our problem."
Friedman argued that the state could renew its arguments to move Kibby's property in a few months, after defense lawyers received enough evidence to argue against it. "Right now this is just unfair; it's repugnant to due process," he said.
"They're being disingenuous," she said. "That's what's repugnant."
Albee said she would rule as soon as possible.


text:
Q: "Object reference not set to an instance of an object" in Powershell Core Im trying to use the AWS .Net SDK in my powershell script. My script works without error in Windows Powershell 5, but whey I try running it in Powershell Core 6, I get the error:

"Object reference not set to an instance of an object."

Add-Type -Path (${env:ProgramFiles(x86)} + "\AWS SDK for .NET\bin\net45\AWSSDK.DynamoDBv2.dll")

$EndPoint = [Amazon.RegionEndpoint]::USWest1

$Client = New-Object -TypeName Amazon.DynamoDBv2.AmazonDynamoDBClient -ArgumentList $EndPoint

$tablename = "foo"

$table = [Amazon.DynamoDBv2.DocumentModel.Table]::LoadTable($Client, $tablename)

I expect the last line to run without error, but in Powershell Core it throws the "Object reference not set to an instance of an object." error

A: This is an old question, and I am guessing you have found the solution to your problem. If so, please share here so we can conclude this question.
For helping other if in case hit this thread, please check https://github.com/aws/aws-tools-for-powershell and understand how to get PowerShell tools to get it working for .net core.

redpajama_set_name: RedPajamaStackExchange


text:
Tag Archives: Wide World of Sports
ESPN Inks New Broadcast Deal With F1
Posted on October 4, 2017 by philspicks
Courtesy: ESPN/ABC/Formula 1
Formula 1 racing is returning to the ESPN networks.
ESPN announced Wednesday the network has reached a broadcast rights deal with Formula 1 to broadcast the FIA Formula 1 Racing Championship series throughout 2018. The deal gives ESPN and ABC the broadcast rights for all 21 of the series' races beginning with the season-opening Formula 1 Australian Grand Prix on March 25 and ending with the Formula 1 Abu Dhabi Grand Prix on Nov. 25.
Burke Magnus, ESPN executive vice president of programming & scheduling, was enthusiastic about the series' return to the ESPN networks after being away for many years.
"ESPN has a long commitment to motorsports, and F1 is a crown jewel in the sport," Magnus said. "There are many passionate Formula 1 fans in the U.S. and we look forward to bringing the pageantry, spectacle and excitement of F1 to viewers across the ESPN platforms."
Sean Bratches, Formula 1 managing director of commercial operations, was just as enthusiastic about the new deal as Magnus.
"We are excited about the return of the world's foremost motor racing platform to the ABC and ESPN platforms," Bratches said. "ABC's Wide World of Sports first started airing live live grands prix in the early 1960s and this linear partnership with ESPN represents a significant step forward in achieving Formula 1's aim of broadening the sport's appeal. The U.S. market is very important growth opportunity for Formula 1 and we are looking forward to working with ESPN to ignite the growing fan interest."
Formula 1 made its Wide World of Sports debut on June 10, 1962 with the Monaco Grand Prix, a week after the race was actually run. ABC ran a handful of Formula 1's races up until 1988. ESPN joined ABC in broadcasting Formula 1 in 1984 with a 10-race schedule.
Following the broadcast of that 10-race schedule, ESPN would continue to air Formula 1 races until 1997, its last year carrying the series. The full schedule for Formula 1's return to the ESPN networks in 2018 is noted below.
2018 FIA Formula One World Championship Schedule on ESPN and ABC
Bahrain Grand Prix
Azerbaijan Grand Prix
ABC *
Austrian Grand Prix
Aug. 26
Russian Grand Prix
United States Grand Prix
Brazilian Grand Prix
* re-air
More information on the network's Formula 1 coverage is available online along with all of Formula 1's latest news at:
Website: http://www.espn.com/racing, http://www.Formula1.com
Facebook: http://www.facebook.com/Formula1
Twitter: http://twitter.com/f1
Posted in Internet, Sports, Television	| Tagged ABC, entertainment, ESPN, F1, facebook, Formula 1, internet, Phil's Picks, Television, Twitter, Wide World of Sports, Wordpress	| Leave a reply
ABC Marking 50th Consecutive Year Broadcasting Indy 500
Posted on May 20, 2014 by philspicks
2014 is a big year for ABC. That's because this year marks the 50th anniversary of the network broadcasting what has become the "Greatest Spectacle in Racing."
ABC started airing highlights of the annual race in 1965 on its then black-and-white Wide World Of Sports telecast. It has now become one of the longest running partnerships between a television network and sporting event. To put things into perspective, CBS has carried weekend coverage of the PGA's Masters Tournament since 1965 as well. And ABC has carried the Little League World Series since 1963.
Allen Bestwick will make his debut anchoring the broadcast this year. He will join a very short list of individuals to ever call the race in its now ninety-eight year history. Only nine others have ever called the race in its rich history. Bestwick said of being tapped to helm this year's broadcast that hearing the broadcast of the Indy 500 was his inspiration for getting into broadcasting when he was younger and that he is rather excited to be helming it this year. "One of the things that sparked my fascination with broadcasting was that appointment viewing of the broadcast of the Indianapolis 500 with Jim McKay behind the microphone," he said. "It's one of those things that attracted me and inspired me to get into the business and to think that I'm going to have the opportunity to sit in that chair – THAT chair – is mind-blowing." Bestwick will be joined for this year's broadcast by analysts Eddie Cheever and Scott Goodyear, both of whom competed in the Indy 500. SportsCenter anchor Lindsay Czarniak will host the broadcast from IMS' iconic Pagoda. Rick DeBruhl, Jamie Little, Vince Welch, and Dr. Jerry Punch will all be in the pits.
This year's festivities are expected to begin with pre-race coverage at 11am ET. The green flag is expected to wave at 12:12pm ET. Viewers can get a whole different angle on the race through ESPN's in-car cameras. The view from those cameras will stream on ESPN3 through WatchESPN and the official Verizon Indycar Series website, http://www.indycar.com. More information on coverage of this year's race is available online at http://espn.go.com/racing and http://www.indycar.com. To keep up with the latest sports and entertainment news and reviews, go online to http://www.facebook.com/philspicks and "Like" it. Fans can always keep up with the latest sports and entertainment news and reviews in the Phil's Picks blog at https://philspicks.wordpress.com.
Posted in Celebrities, Internet, Sports, Television	| Tagged ABC, Allen Bestwick, celebrities, Dr. Jerry Punch, Eddie Cheever, entertainment, ESPN, ESPN3, facebook, Indianapolis Motor Speedway, Indy 500, internet, Jamie Little, Jim McKay, Lindsay Czarniak, Phils Picks, Rick DeBruhl, Scott Goodyear, sports, SportsCenter, Television, Verizon, Verizon IndyCar Series, WatchESPN, Wide World of Sports, Wordpress	| Leave a reply


A UNESCO committee omitted Australia's Great Barrier Reef from its list of world heritage sites it considers to be in danger, a move the government welcomed but conservationists questioned.
Bushfires engulfing Australian Tasmania could have detrimental long-term effects on the state's unique world heritage forests, according to conservationists.
Water level in the Lake Baikal declined to the extreme low limit of 456 meters [under the Pacific Elevation System] and dropped 0.01 m below it last weekend, Russian water resources authority said on Monday on its website.
The water level at Lake Baikal, the world's largest freshwater source, having dropped to critical marks in April, may fall by another 28 centimetres next spring, the head of the Baikal Institute told TASS on Tuesday.
Following the massive destruction to world heritage monuments in Nepal after Saturdays devastating earthquake, UNESCO on Monday said they would field an international expert mission to undertake an in-depth damage assessment of the damage to heritage and path to recovery.
LHASA, Aug. 13 (Xinhua) -- The Chinese government's efforts to preserve intangible heritage in Tibet and other ethnic regions have proven effective, said sources at the first Tibet development forum Wednesday.
"Of the 386 million yuan (62.7 million U.S. dollars) for cultural heritage preservation allocated by the central treasury between 2002 and 2009, about a quarter was spent in ethnic regions including Tibet," said Li Jianhui, a culture promotion official with the State Ethnic Affairs Commission.
Australia's Great Barrier Reef remains under threat despite efforts to rein in major sources of damage to the World Heritage-listed icon, the government said on Tuesday.
Canberra released a five-yearly review of the reef and moves to protect it, to address concerns raised by UNESCO and persuade the world body not to put the key tourist attraction on its "in danger" list next year. "Even with the recent management initiatives to reduce threats and improve resilience, the overall outlook for the Great Barrier Reef is poor, has worsened since 2009 and is expected to further deteriorate," the government said in its outlook report.
Concerns have been raised over the protection of historic towns as persistent downpours in Central and Southwest China have flooded Fenghuang, a renowned tourism destination in Hunan Province.
More than 120,000 people were relocated amid power cuts in Fenghuang on Wednesday.
China gained two more UNESCO World Heritage sites Sunday, bringing the nation's total to 47.
The decision was announced by the UNESCO World Heritage Committee in Qatar.
The Silk Road, which served as a corridor for trade and cultural exchanges between Asia and Europe dating back 2,000 years ago, was inscribed on the World Heritage List along with the Grand Canal, which runs 1,794 kilometers from Beijing to Hangzhou, East China's Zhejiang Province.
Australia hails Barrier Reef deferral. Environmentalists call UNESCO decision "final warning"
Australia Thursday called a decision by UNESCO to defer listing the Great Barrier Reef as in danger "a win for logic," but environmentalists said it was a final warning.
The UN cultural agency on Wednesday said the reef could be put on a list of endangered World Heritage Sites if more was not done to protect it.
It voiced alarm at a "serious decline in the condition" of the reef, and said "a business as usual approach to managing the property is not an option."
Australia was given until February 1 next year to submit a report on what it was doing to protect the natural wonder. The Queensland state government saw the deferral as "a tick of approval."
The United Nations world heritage body UNESCO will decide on Wednesday whether to list Australia's Great Barrier Reef as "in danger."
Concerns were raised in May over the decision to allow dredging near the reef, the ABC reported.
UNESCO recommended adding the reef to the World Heritage in Danger list in 2015, unless the Queensland Government took further action to protect it.
The committee is meeting this week in Doha and Queensland Environment Minister Andrew Powell is also there to convince UNESCO the reef is not in danger.
Silk Road inscription into World Heritage list "hopeful"
The multinational campaign to include the Silk Road into the World Heritage list has entered the final countdown, with officials and experts sanguine about its success.
Jointly submitted by China, Kazakhstan and Kyrgyzstan, the application for adding part of the millennium-old trade route into the UNESCO list is expected to be finalized by a vote in June, when the World Heritage Committee convenes its 38th session in Doha.


text:
Honours in Sociology or Social Anthropology
The Honours programme offers students a firm foundation in the theoretical and methodological underpinnings of each of the two disciplines, on which further postgraduate work can be built.
Students interested in careers as academics, researchers, social and policy analysts, planners, developers, consultants, social workers, human resource managers, and communications, government and NGO officials will all benefit, as will individuals with a general interest in deepening and broadening their understanding of society and social dynamics.
The programme aims:
To develop theoretical insight concerning social relations, social institutions, and the dynamics of social change;
To develop rigorous conceptual and research skills;
To deepen understanding of social dynamics within the southern African region using a comparative perspective; and
To encourage students to develop their individual areas of research interest and expertise.
The programme is designed as a residential programme for full-time study, involving regular class meetings and seminars as well as time for individual reading and research. It takes a minimum of a year to complete. Research projects are usually handed in for examination by mid-December, meaning that successful students will graduate at the following March graduation ceremony.
The two disciplines run in parallel with integration of classes at particular points. Students obtain either the degree BA Honours in Sociology or the degree BA Honours in Social Anthropology depending on their undergraduate major and their selected area of specialisation.
The Honours programme consists of an induction programme and four modules. The induction programme may involve a group exercise such as a departmental field trip as well as various orientation sessions.
Two compulsory modules cater for the theoretical and methodological foundations of the relevant disciplines, i.e. Sociology and Social Anthropology. A third compulsory component consists of the research project which is conducted under departmental supervision.
The fourth compulsory component comprises a module on "Selected themes" that is divided into two components. The first component addresses the disciplinary foci of within Sociology and Social Anthropology, respectively. This half-module will be presented in the
3rd term, primarily as separate sessions for Sociology and Social Anthropology students.
Finally, for the fourth term, students may choose one of a number of elective half-modules dealing with contemporary, relevant issues and themes in Sociology and/or Social Anthropology. The offering may vary from year to year.
In summary, the programme looks as follows:
Module 1: Sociological/Social Anthropological theory;
Module 2: Sociological/Social Anthropological research;
Module 3: Research project, under supervision;
Module 4: Selected themes:
– Disciplinary foci; and
– Elective. Students choose one theme from areas such as: development studies; the sociology of work and innovation; HIV/AIDS and society; land, environment and society; culture, identity and gender; and religion.
Assessment is continuous, involving essays, seminar participation and examinations and students obtain a mark for every module completed. Students are also expected to participate actively in the intellectual life of the Department through, for instance, attendance of the weekly departmental seminar.
In cases where a student"s average for the programme as a whole is borderline in terms of the class of the degree (for instance, the student has an average of 74%, just short of the 75% required for a cum laude pass) the Department may request the student to participate in an oral examination to determine the final mark.
The research project
The research project is examined by an internal examiner and moderated externally. Students choose their topics in consultation with the Department and work under the supervision of a member of the academic staff. The final 1.5 spaced project report is expected to be in the region of 10 000 – 12 000 words (maximum 15 000 words), excluding references.
For more information, please contact one of the programme co-ordinators:
Programme co-ordinators
Social Anthropology:
Dr Ilana van Wyk
E-mail: ilanavw@sun.ac.za
Sociology:
Dr Khayaat Fakier
E-mail: kfakier@sun.ac.za
FUNDING OPPORTUNITIES AT THE DEPARTMENT OF SOCIOLOGY AND SOCIAL ANTHROPOLOGY
For the 2017 Honours Year, the department is offering 14 Scholarships.
Six are Mellon Foundation Critical Transformation Studies Scholarships
Four are SP Cilliers Bursaries
Four are linked to the SARCHI chair in Land, Environment and Society.
To apply for these scholarships, you need to send an application letter the length of a single typed page that offers your motivation for doing honours and makes your specific interests in the programme clear. You may also choose to include a statement of your financial need, and you may be asked to furnish your financial details during the selection process.
The criteria for these awards are a combination of merit, historical disadvantage and financial need.
Application to these scholarships is separate from your application to honours.
Application letters need to be sent to Genay Dhelminie (genay@sun.ac.za) and must reach her by 30 November 2016.
Applications to Mellon and SP Cilliers will be considered together. One application is sufficient for both.
Applications to the SARCHI Chair in Land, Environment and Society will be considered separately and be sure to indicate that you are applying for this award in your application letter. Please make clear in this interest in this broad research area
Awards will be announced by the end of the second week of December.
EXTRA-DEPARTMENTAL FUNDING OPPORTUNITIES
In addition to this funding, there are other sources of funding you may wish to explore.
There are some National Research Foundation Bursaries that are external to the University, but about which you should consult the Postgraduate and International Office.
There are University level bursaries administered by the Postgraduate and International office, including Academic Merit Bursaries.


text:
\section{Introduction}\label{sec:Intro}
It has been an open question for forty years as to whether the smallest known universal Turing machines (UTMs) are efficient simulators of Turing machines. This question is intimately related to a problem regarding the computational complexity of 2-tag systems.

Shannon~\cite{Sha1956p} was the first to consider the question of finding the smallest possible UTM, where size is the number of states and symbols. Early attempts~\cite{Ike1958x,Wat1961p} gave small UTMs that efficiently (in polynomial time) simulate Turing machines.

In the early 1960s Cocke and Minsky~\cite{CM1964p} showed that 2-tag systems simulate Turing machines, but in an exponentially slow fashion. Minsky~\cite{Min1962p} found a small 7-state, 4-symbol UTM that simulates 2-tag systems in polynomial time. So this small UTM simulates Turing machines via the following sequence of simulations
\begin{equation}\label{eq:MinskyRogozhinSim}
\textrm{Turing machine}\simulates \textrm{2-tag system} \simulates \textrm{small UTM}
\end{equation}
where $A\simulates B$ denotes that $A$ is simulated by $B$. Later Rogozhin~\cite{Rog1996p} and others~\cite{Bai2001c,KR2001c,Rob1991p} used Minsky's technique of simulation via~\eqref{eq:MinskyRogozhinSim} to find small UTMs for a range of state-symbol pairs (see Figure~\ref{fig:PECurve-april06}). All of these small UTMs efficiently simulate 2-tag systems. However since the 2-tag simulation of Turing machines is exponentially slow it has remained an open problem as to whether these UTMs can be made to run in polynomial time.

\begin{figure}
\begin{center}
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\psset{unit=2.4ex}

\begin{pspicture}(0,-3)(22,20)
\psset{dotsize=5pt}
\psset{dotstyle=square*}
\psdot (17.6,19.9)

\put (17.2,19.0) {\footnotesize explicitly constructed}
\put (17.2,18.2) {\footnotesize polynomial time UTMs}

\psset{linestyle=solid}
\psset{linecolor=white}
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\psset{linecolor=black}
\pscircle(17.6,16.7){0.25}
\put (17.2,15.8) {\footnotesize  explicitly constructed UTMs}
\put (17.2,15.0) {\footnotesize  of Minsky, Rogozhin, et al.}

\psset{linestyle=dotted}
\psline (17.2,13.5)(23,13.5)
\put (17.2,12.7) {\footnotesize  previous polynomial}
\put (17.2,11.9) {\footnotesize  time curve}

\psset{linestyle=dashed, dash=3pt 3pt}
\psline (17.2,10.4)(23,10.4)
\put (17.2,9.6) {\footnotesize  previous exponential}
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\psset{linestyle=solid}
\psline (17.2,6.5)(23,6.5)
\put (17.2,5.7) {\footnotesize  non-universal curve}

\psset{linestyle=solid}

{\tiny \psaxes[ticksize=1pt]{->}(20,19) }
\rput (5,-1.8) {\footnotesize states}
\put (-4.5,3.3) {\footnotesize symbols}

\psline (3,1)(20,1)
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\psline (3,1)(3,2)

\psset{dotstyle=square*}

\psdot (3,11)
\psdot (5,7)
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\psdot (8,4)

\psset{linestyle=dotted}
\psline (3,11)(3,19) %
\psline (3,11)(5,11)
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\psline (6,6)(7,6)
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\psline (8,4)(20,4)

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\psset{linestyle=solid}
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\psset{linestyle=solid}
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\qdisk(19,2){0.25}
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\pscircle(19,2){0.25}
\end{pspicture}
\caption{State-symbol plot of small UTMs. The plot shows the polynomial time curve induced by our previous UTMs~\cite{NW2006p--acc}, the exponential time curve of Minsky, Rogozhin and others~\cite{Bai2001c,KR2001c,Min1962p,Rog1996p}, and the non-universal %
curve for which there are no UTMs~\cite{Her1968c,Kud1996p,Pav1973p,Pav1978p}. The present paper improves the polynomial time curve so that it coincides with the previous exponential time curve. Our result shows that a polynomial time UTM exists for each state-symbol pair that is on, above, and to the right of the new polynomial time curve.}\label{fig:PECurve-april06}
\end{center}
\end{figure}

For our result we replace~\eqref{eq:MinskyRogozhinSim} with the following sequence of simulations
\begin{align*}%
& \textrm{Turing machine} \simulates  \textrm{cyclic tag system}\\
& \simulates \textrm{2-tag system} \simulates \textrm{small UTM} %
\end{align*}
Specifically, in this paper we show that 2-tag systems efficiently simulate cyclic tag systems. In a recent paper~\cite{NW2006c} we have shown that cyclic tag systems efficiently simulate Turing machines. Thus the present paper provides an important piece of the puzzle as far as the computational complexity of small UTMs and 2-tag systems are concerned. Our main results states:
\begin{theorem}\label{thm:2-tagSimTM}
Given a single tape deterministic Turing machine $M$ that computes in time~$t$ then there is a 2-tag system~$T_M$ that simulates the computation of~$M$ and computes in polynomial time $O(t^{4}(\log t)^{2})$.
\end{theorem}
This immediately gives the following interesting result.
\begin{corollary}\label{cor:UTMsSimTM}
The small UTMs of Minsky, Rogozhin and others~\cite{Bai2001c,KR2001c,Min1962p,Rob1991p,Rog1996p} are polynomial time, $O(t^{8}(\log t)^{4})$, simulators of Turing machines.
\end{corollary}

Before our result it was entirely plausible that there was an exponential trade-off between UTM program size complexity, and time/space complexity; the smallest UTMs seemed to be exponentially slow. However our result shows there is currently little evidence for such a claim.

Early examples of efficient small UTMs were found by Ikeno and Watanabe~\cite{Ike1958x,Wat1961p}.  Prior to the present paper the smallest known polynomial time UTMs were to be found in~\cite{NW2006p--acc}. However these efficient machines are not as small as those of Rogozhin et al., hence the present paper represents a significant size improvement when considering small polynomial time UTMs. This improvement is illustrated in Figure~\ref{fig:PECurve-april06}.

There are numerous other applications of Theorem~\ref{thm:2-tagSimTM}. The technique of simulation via 2-tag systems is at the core of many results in the broad survey by Margenstern~\cite{Mar2000p}; our result exponentially improves the time overheads in many~\cite{LN1990p,Mar1993c,Mar1995c,Mar1995bc,Rob1971p} of these constructions.
Another example of the use of small efficient UTMs is given by  Levin and Venkatesan~\cite{LV2001m,VL1988c} who used the small 8-state, 5-symbol, polynomial time UTM of Watanabe's~\cite{Wat1961p} to show the average case NP-completeness of a graph colouring problem. %

In the present paper, the phrase ``small UTMs'' refers to Turing machines that obey the standard definitions. Recently Cook~\cite{Coo2004p} has found universal machines that are significantly smaller than those discussed in the present paper. Cook's machines simulate the cellular automaton Rule 110, which Cook showed to be universal via an impressive simulation. However these machines are generalisations of standard Turing machines: their blank tape consists of an infinitely repeated word to the left and another to the right. Intuitively, this change of definition makes quite a difference, especially since Cook encodes a (possibly universal) program in one of these repeated words. It should be noted that Cook's UTMs are exponentially slow; in a recent paper~\cite{NW2006c} we have improved their simulation time overhead to polynomial.

\subsection{Preliminaries}
Let $\Nset=\{0,1,2,\ldots\}$, $\Sigma$ denote a finite alphabet, and~$\epsilon$ denote the empty word. The function $\Vert\,\Vert : \Sigma^{\ast}\times\Sigma^{\ast} \rightarrow \Nset$ is written as $\Vert w_1,w_2 \Vert$ and gives the number of occurrences of the word~$w_2$ in~$w_1$. For example $\Vert 1101001,01 \Vert =2$. All logs are in base~2. Throughout the paper~$a$ and~$\#$ are constant symbols, and $x,x_i \in\{0,1\}$.
We define the {\em parity} of a word to be {\em odd} if the read (leftmost) symbol is undotted and {\em even} if the read symbol is dotted.

\section{2-tag systems}\label{sec:2tagDef}
Tag systems where introducted by Post~\cite{Pos1943p}, and 2-tag systems were shown to be universal by Cocke and Minsky~\cite{CM1964p}.
\begin{definition}[2-tag system]\label{def:2TS}
A tag system consists of a finite alphabet of symbols $\Sigma$, a finite set of rules $R : \Sigma\rightarrow\Sigma^{\ast}$ and a deletion number $\beta\in\Nset$, $\beta\geqslant 1$. For a 2-tag system $\beta = 2$.
\end{definition}
The 2-tag systems we consider are deterministic (or equivalently, monogenic~\cite{Coo1966p,Wan1963p}). The computation of a 2-tag system acts on a dataword $w=\sigma_1 \sigma_2\ldots \sigma_l$. The entire configuration is given by $w$. In a computation step, the symbols~$\sigma_1 \sigma_2$ are deleted and if there is a rule for $\sigma_1$, i.e.\ a rule of the form $\sigma_1\rightarrow \sigma_{l+1}\ldots \sigma_{l+c}$, then the word $\sigma_{l+1}\ldots \sigma_{l+c}$ is appended. We write $w_1\vdash w_2$ when the dataword (configuration) $w_2$ is obtained from $w_1$ via a single computation step:
\begin{equation*}
\sigma_1 \sigma_2 \sigma_3\ldots \sigma_l \vdash \sigma_3\ldots \sigma_l \sigma_{l+1}\ldots \sigma_{l+c}
\end{equation*}
where $\sigma_1\rightarrow \sigma_{l+1}\ldots \sigma_{l+c} \in R$.\hspace{0.2ex} A 2-tag system completes its computation if~(i)~$|w|< \beta $, or (ii)~it enters a repeating sequence of configurations, or (iii)~there is no rule for the leftmost symbol.
The complexity measures of time and space are defined in the obvious way. Given a word~$w$, we use the term {\em round} to describe~$\lceil \vert w\vert/2\rceil$ computation steps that traverse~$w$ exactly once~\cite{Coo1966p,Wan1963p}.

We often write 2-tag symbols in pairs. The second (even numbered) symbol is dotted to distinguish it from the first. In the sequel we encode binary symbols in the following way, $1$ is encoded as $\oes{1}{{}}{{}}$ and 0 as $\oes{0}{{}}{{}}$.  Also a single pair of symbols is distinguished by being `barred': $\osr{0}{{}}{\,}\esr{0}{{}}{\,}$ or $\osr{1}{{}}{\,}\esr{1}{{}}{\,}$.
So an example encoding of the word~$11010$ is~$\osr{1}{{}}{{}}\esr{1}{{}}{{}}\oes{1}{{}}{{}}\oes{0}{{}}{{}}\oes{1}{{}}{{}}\oes{0}{{}}{{}}$.
\begin{lemma}\label{lem:oddEven}Let
 $w=\osr{x}{0}{{}}\esr{x}{0}{{}}x_1x_2\ldots x_l \in\{\osr{0}{{}}{{}}\esr{0}{{}}{{}},\osr{1}{{}}{{}}\esr{1}{{}}{{}}\}\{0,1\}^{\ast}$. Then there is a 2-tag system $T$ that tests whether~$\vert  w\vert $ is odd or even in exactly~$\lfloor\vert w\vert /2\rfloor+1$ timesteps.\end{lemma}
\begin{proof}
The 2-tag system has 6 rules, $R=\{\osr{x}{{}}{{}}\rightarrow \osr{x}{{}}{1}\esr{x}{{}}{1}, x\rightarrow \epsilon,\osr{x}{{}}{1}\rightarrow\os{x}{{}}{2}\}$ where $x\in\{0,1\}$. Initially $T$ reads the leftmost symbol $\osr{x}{0}{{}}$ of $w$. After one round, if the read symbol is dotted then the output is the single symbol $\esr{x}{{}}{1}$ signifying that~$|w|$ is odd. Otherwise the output is $\os{x}{{}}{2}$ signifying that~$|w|$ is even.\qed \end{proof}

Despite its simplicity, the proof idea of Lemma~\ref{lem:oddEven} constitutes one of the main ingredients of Cocke and Minsky's (exponentially slow) 2-tag simulation of Turing machines~\cite{CM1964p}. The 2-tag system dataword encodes an arbitrary TM configuration as two unary numbers. The left side of the tape is encoded as one unary number, the right side as another. Their simulation makes use of repeated tests for oddness and evenness of dataword length. Also doubling and halving of dataword length is used to read and write to the simulated tape.

\section{Cyclic tag systems}
Cyclic tag systems were introduced by Cook~\cite{Coo2004p}.
\begin{definition}[cyclic tag system~\cite{Coo2004p}]\label{def:CyclicTS}
A cyclic tag system $C=\alpha_{0},\alpha_{1},\ldots,\alpha_{p-1}$ is a list of binary words $\alpha_{\arbApp}\in\{0,1\}^{\ast}$ called appendants.
\end{definition}
A {\em configuration} of a cyclic tag system consists of (i) a {\em marker} that points to a single appendant $\alpha_{\arbApp}$ in $C$, and (ii) a dataword $w = x_0 x_1\ldots x_l \in\{0,1\}^*$. Intuitively the list $C$ is a {\em program} with the marker pointing to instruction $\alpha_{\arbApp}$. At the initial configuration the marker points to appendant $\alpha_0$ and $w$ is the binary input word.
\begin{definition}[computation step of a cyclic tag system]\label{def:OperationJTS}
A computation step is deterministic and acts on a configuration in one of two ways:
\begin{itemize}
\item If $x_0 = 0$ then $x_0$ is deleted and the marker moves to appendant $\alpha_{({\arbApp}+1)\mod p}$.
\item If $x_0 = 1$ then $x_0$ is deleted, the word $\alpha_{\arbApp}$ is appended onto the right end of $w$,  and the marker moves to appendant $\alpha_{({\arbApp}+1)\mod p}$.
\end{itemize}
\end{definition}

A cyclic tag system completes its computation if (i) the dataword is the empty word, or (ii) it enters a repeating sequence of configurations. The complexity measures of time and space are defined in the obvious way.

\begin{example}
\textit{(cyclic tag system computation)} Let $C=00,010,11$ be a cyclic tag system with input word $011$. Below we give the first four steps of the computation. In each configuration $C$ is given on the left with the marked appendant highlighted in bold font.
\begin{eqnarray*}
\begin{aligned}
            & \pmb{00},010,11 \quad 011 & \vdash\quad  & 00,\pmb{010},11 \quad 11 & \\
\vdash\quad & 00,010,\pmb{11} \quad 1010 & \vdash\quad  & \pmb{00},010,11 \quad 01011 & \\
\vdash\quad & 00,\pmb{010},11 \quad 1011 & \vdash\quad  &\ldots
\end{aligned}
\end{eqnarray*}
\end{example}
We write an arbitrary single step of a cyclic tag system computation as
\begin{equation}\label{eq:cyclicTagStepToBeSim}
\begin{aligned} %
& \alpha_0
 ,\ldots,
 \alpha_{{\arbApp}-1},
 \pmb{\alpha_{{\arbApp}}},
 \alpha_{{\arbApp}+1},\ldots,\alpha_{p-1} \quad x_0 x_1 \ldots x_l \\
\vdash\quad   &
\alpha_0,
\ldots,
\alpha_{{\arbApp}},
\pmb{\alpha_{{\arbApp}+1}},
\alpha_{{\arbApp}+2},
\ldots,
\alpha_{p-1} \quad  x_1\!\ldots x_l x_{l+1}\!\ldots x_{l+c}
\end{aligned} %
\end{equation}
where $x\in\{0,1\}$, and as usual if $x_0 = 0$ then $x_{l+1} \ldots x_{l+c}=\epsilon$, otherwise if~$x_0 = 1$ then $x_{l+1} \ldots x_{l+c} = \alpha_{\arbApp}\in\{0,1\}^{c}, c\in\Nset$.

Cook~\cite{Coo2004p} used the universality of cyclic tag systems to show that Rule~110, a binary one-dimensional cellular automaton, is universal. Recently we have improved on Cook's work by showing that cyclic tag systems simulate Turing machines in polynomial time:
\begin{theorem}[\cite{NW2006c}]\label{lem:CycTagSysSimTM}
Let $M$ be a single-tape deterministic Turing machine that computes in time~$t$. Then there is a cyclic tag system $C_M$ that simulates the computation of $M$ in time $O(t^{3} \log t)$.
\end{theorem}
In order to calculate this upper bound we substitute space bounds for time bounds whenever possible in the analysis.

\section{2-tag systems efficiently simulate cyclic tag systems}
In this section we prove Theorem~\ref{thm:2-tagSimTM}.

\subsection{Encoding}
Cyclic tag systems use a binary alphabet and program control is determined by the read symbol and the value of the program instruction marker. On the one hand 2-tag systems seem more general than cyclic tag systems as an arbitrary constant (independent of input length) sized alphabet is permitted. On the other hand 2-tag systems seem more restricted in that program control is determined solely by the read symbol.

Because of this restriction we use a large number of symbols in our construction. The number of such symbols is a constant that is independent of input length, but is dependent on our simulation algorithm and the size of the simulated cyclic tag system program.  In our encoding we decorate symbols with dots ($\es{x}{{}}{{}},\thirdsymbol{x}{{}}{{}}$), bars (\osr{x}{{}}{{}}) and under-indexes~$(\os{x}{{}}{j})$. These decorations are used for algorithm control flow.

\begin{definition}[2-tag input encoding]\label{def:2tagInputEnc} The cyclic tag system input dataword $w=x_0x_1\ldots x_n\in\{0,1\}^{\ast}$ is encoded as the 2-tag dataword
\begin{equation*}
\hat{w}=
\oesr{x}{0}{1}\;
\oes{x}{1}{1}\;
\oes{x}{2}{1}\ldots
\oes{x}{n}{1}\;
\oes{a}{{}}{1}\; \oes{a}{{}}{1} \ldots \oes{a}{{}}{1}
\end{equation*}
where the number of $\oes{a}{{}}{\,}$ pairs in $\hat{w}$ is $\Vert\hat{w},\oes{a}{{}}{\,}\Vert=2^{\lceil\log (n+1)\rceil}$ and  the extra whitespace between symbol pairs is for human readability purposes only.
\end{definition}

This encoding is computable in logspace. The subword $\oes{a}{{}}{\,}\,\oes{a}{{}}{\,}\ldots\oes{a}{{}}{\,}$ is used as a {\em counter} and its {\em value} $\Vert\hat{w},\oes{a}{{}}{\,}\Vert$ is used extensively in our algorithms below.

An arbitrary (not necessarily input) cyclic dataword is encoded similarly to Definition~\ref{def:2tagInputEnc} except that the counter is `embedded' in $w$. Specifically if $w=x_0x_1\ldots x_l$ then
\begin{equation}\label{eq:2tagArbitraryWordEnc}
\hat{w}=
\oesr{x}{0}{j}\; \oes{x}{1}{j}\ldots\oes{x}{{i}}{j}\;
\oes{a}{{}}{j}\ldots\oes{a}{{}}{j}\;
\oes{x}{{i+1}}{j}\ldots\oes{x}{{l}}{j}
\end{equation}
for some $i\in\{0,\ldots,l\}$ and $j\in\Nset$. Furthermore the counter value is some power of 2 strictly greater than $l$, that is
\begin{equation}\label{eq:counterValue}
\Vert\hat{w},\oes{a}{{}}{\,}\Vert = 2^{\lceil\log l'\rceil}
\end{equation}
where $l'\in\Nset$, $l' > l$.

In our simulation the counter does not grow too large. As will be shown, the counter value is never more than double the longest dataword length that occurs in the simulated cyclic tag system computation.

\subsection{The simulation}\label{sec:sim}
We wish to show that there is a 2-tag system that simulates an arbitrary single step of a cyclic tag system computation, as defined in Equation~\eqref{eq:cyclicTagStepToBeSim}. We decompose Equation~\eqref{eq:cyclicTagStepToBeSim} into three conceptual steps: (i)~if $x_0 = 1$ then simulate the rule $x_0\rightarrow \alpha_{\arbApp}$ by appending $\alpha_{\arbApp} = x_{l+1} \ldots x_{l+c}$, (ii)~set $x_1$ to be the new read symbol and delete $x_0$, (iii)~increment the program marker ${\arbApp}$ so that the next appendant is $\alpha_{({\arbApp}+1 \mod p)}$.

We begin by giving a simulation of (ii). Informally speaking, Lemma~\ref{lem:passBar} states that there is a 2-tag system that (efficiently) moves the `bar' forward by one symbol pair.  The main difficulty is in distinguishing $\oes{x}{1}{1}$ from the other unbarred symbol pairs $\oes{x}{2}{2}, \ldots, \oes{x}{l}{l}$. This Lemma is stated for the case that the counter value is the next power of two greater than the encoded dataword length.

\begin{lemma}\label{lem:passBar}
Given a word of the form
\begin{equation*}
\hat{w}=
\oesr{x}{0}{1}\;
\oes{x}{1}{1}\ldots\oes{x}{{i}}{1}\;
\oes{a}{{}}{1}\ldots\oes{a}{{}}{1}\;
\oes{x}{{i+1}}{1}\ldots\oes{x}{{l}}{1}
\end{equation*}
 then there is a 2-tag system $T$ that computes
\begin{equation*}
\hathatw =
\oesr{x}{1}{6}\ldots\oes{x}{{i}}{6}\;
\oes{a}{{}}{6}  %
\ldots\oes{a}{{}}{6}\;
\oes{x}{{i+1}}{6} %
\ldots\oes{x}{{l}}{6}
\end{equation*}
in time $O(l \log l)$. Here $\Vert\hat{w},\oes{a}{{}}{\,}\Vert= \Vert\hathatw ,\oes{a}{{}}{\,}\Vert =2^{\lceil\log (l+1)\rceil}$ and $i \in\{0,\ldots,l\}$.
\end{lemma}
\begin{proof}[Proof idea] There are 5 stages to the 2-tag algorithm. We let $\hat{w}_0 = \hat{w}$ and let~$\hat{w}_k$ denote the output of the $k^{\textrm{th}}$ iteration of the 5 stages.

In Stages 1 to 3 of iteration $k$ we compute $\lceil\Vert \hat{w}_{k-1},\oes{x}{{}}{{}} \Vert /2 \rceil$, by marking every second pair of $\oes{x}{{}}{{}}$ symbols (we mark the even numbered pairs). Then in Stages 4 and 5 we halve $\Vert \hat{w}_{k-1},\oes{a}{{}}{{}} \Vert$ by marking every second pair of~$\oes{a}{{}}{{}}$ symbols (again we mark the even numbered pairs). We then return to Stage 1 and iterate until $\Vert \hat{w}_k,\oes{a}{{}}{{}} \Vert=1$: the counter has an odd value for the first time and we detect this. The number of fully completed iterations, and final value for~$k$, is~$k=\log \Vert \hat{w},\oes{a}{{}}{{}} \Vert$. At this point~$\oes{x}{1}{{}}$ is the only pair of unmarked~$\oes{x}{{}}{{}}$ symbols in~$\hat{w}$, and so~$\oes{x}{1}{{}}$ is isolated (unique) from the other symbol pairs in~$\hat{w}$.
We delete~$\osr{x}{0}{{}}\esr{x}{0}{{}}$. The uniqueness of $\oes{x}{1}{{}}$ enables the rule $\os{x}{1}{{}}\rightarrow\osr{x}{1}{{}}\esr{x}{1}{{}}$ to be executed successfully.
\end{proof}
\begin{proof}[Proof details]
As usual let $x\in\{0,1\}$. Here we specify 2-tag rules, and take the reader through a single (the first) iteration of these rules, for illustration purposes we choose~$i=l$.

In Stage 1 we begin with a word of the form
\begin{equation*}
\oesr{x}{0}{1}\;
\oes{x}{1}{1}\;
\oes{x}{2}{1}
\ldots
\oes{x}{l}{1}\;
\oes{a}{{}}{1}\; \oes{a}{{}}{1} \ldots \oes{a}{{}}{1}
\end{equation*}
Stage 1 consists of the rules:
\begin{equation*}%
\{\dwrule{\osr{x}{{}}{1}}{\osr{x}{{}}{2}\esr{x}{{}}{2}},
\dwrule{\os{x}{{}}{1}}{\threesymb{x}{{}}{2}},
\dwrule{\osm{x}{{}}{1}}{\oesm{x}{{}}{2}},
\dwrule{\os{a}{{}}{1}}{\oes{a}{{}}{2}},
\dwrule{\osm{a}{{}}{1}}{\oesm{a}{{}}{2}}\}
\end{equation*}
as well as a few more rules that are given below. After one round we have
\begin{equation}\label{eq:PassBarSimStage1}
\oesr{x}{0}{2}\;
\threesymb{x}{1}{2}\;
\threesymb{x}{2}{2}%
\ldots
\threesymb{x}{l}{2}\;
\oes{a}{{}}{2}\; \oes{a}{{}}{2} \ldots \oes{a}{{}}{2}
\end{equation}
The Stage 2 rules are
\begin{align*}%
\{
& \dwrule{\osr{x}{{}}{2}}{\osr{x}{{}}{3}\esr{x}{{}}{3}},
\dwrule{\os{x}{{}}{2}}{\oes{x}{{}}{3}},
\dwrule{\es{x}{{}}{2}}{\oesm{x}{{}}{3}},
\dwrule{\thirdsymbol{x}{{}}{2}}{\epsilon},
\dwrule{\osm{x}{{}}{2}}{\oesm{x}{{}}{3}},
\dwrule{\esm{x}{{}}{2}}{\oesm{x}{{}}{3}},\\
& \dwrule{\os{a}{{}}{2}}{\oes{a}{{}}{3}},
\dwrule{\es{a}{{}}{2}}{\oes{a}{{}}{3}},
\dwrule{\osm{a}{{}}{2}}{\oesm{a}{{}}{3}},
\dwrule{\esm{a}{{}}{2}}{\oesm{a}{{}}{3}}
\}
\end{align*}
Continuing from~\eqref{eq:PassBarSimStage1}, after one round we see that every second (even numbered) pair of $\oes{x}{{}}{{}}$ is marked
\begin{equation*}%
\oesr{x}{0}{3}\;
\oes{x}{1}{3}\;
\oesm{x}{2}{3}\;
\oes{x}{3}{3}\;
\oesm{x}{4}{3}%
\ldots
\oes{x}{{l-1}}{3}\;
\oesm{x}{l}{3}\;
\oes{a}{{}}{3}\;
\oes{a}{{}}{3}
\ldots
\oes{a}{{}}{3}
\end{equation*}
where (for illustration purposes only) we assume that $l$ is even.

The Stage 3 rules are:
\begin{align*}%
\{
& \dwrule{\osr{x}{{}}{3}}{\osr{x}{{}}{4}\esr{x}{{}}{4}},
\dwrule{\os{x}{{}}{3}}{\oes{x}{{}}{4}},
\dwrule{\osm{x}{{}}{3}}{\oesm{x}{{}}{4}},
\dwrule{\os{a}{{}}{3}}{\oes{a}{{}}{4}},
\dwrule{\osm{a}{{}}{3}}{\oesm{a}{{}}{4}}, \\
& \dwrule{\esr{x}{{}}{3}}{\#\osr{x}{{}}{{4}}\esr{x}{{}}{{4}}},
\dwrule{\es{x}{{}}{3}}{\oes{x}{{}}{4}},
\dwrule{\esm{x}{{}}{3}}{\oesm{x}{{}}{4}},
\dwrule{\es{a}{{}}{3}}{\oes{a}{{}}{4}},
\dwrule{\esm{a}{{}}{3}}{\oesm{a}{{}}{4}}
\}
\end{align*}
We enter Stage 3 by reading either a dotted symbol (there was an odd number of unmarked $\oes{x}{{}}{{}}$ pairs in Stage 1) or undotted  symbol (there was an even number of unmarked $\oes{x}{{}}{{}}$ pairs in Stage 1). Stage 3 begins by checking this (in one step); if the parity is even, that is the 2-tag system is reading dotted symbols, then a~$\#$ symbol is appended to restore the parity to odd after one round.
On completion of Stage 3 we are reading an undotted symbol:
\begin{equation*}%
\oesr{x}{0}{4}\;
\oes{x}{1}{4}\;
\oesm{x}{2}{4}\;
\oes{x}{3}{4}\;
\oesm{x}{4}{4}
\ldots
\oes{x}{{l-1}}{4}\;
\oesm{x}{l}{4}\;
\oes{a}{{}}{4}\; \oes{a}{{}}{4} \ldots \oes{a}{{}}{4}
\end{equation*}

In Stages 4 and 5 of iteration $k$ we halve the value of the counter (we compute $\Vert\hat{w}_{k},\oes{a}{{}}{\,}\Vert = \Vert\hat{w}_{k-1},\oes{a}{{}}{\,}\Vert /2$), in a similar fashion to Stages 1 to 3. The Stage 4 rules are
\begin{equation*}%
\{
 \dwrule{\osr{x}{{}}{4}}{\osr{x}{{}}{5}\esr{x}{{}}{5}},
\dwrule{\os{x}{{}}{4}}{\oes{x}{{}}{5}},
\dwrule{\osm{x}{{}}{4}}{\oesm{x}{{}}{5}},
 \dwrule{\os{a}{{}}{4}}{\threesymb{a}{{}}{5}},
\dwrule{\osm{a}{{}}{4}}{\oesm{a}{{}}{5}}
\}
\end{equation*}
Which after one round gives
\begin{equation*}%
\oesr{x}{0}{5}\;
\oes{x}{1}{5}\;
\oesm{x}{2}{5}
\ldots
\oesm{x}{l}{5}\;
\threesymb{a}{{}}{5}\; \threesymb{a}{{}}{5} \ldots \threesymb{a}{{}}{5}
\end{equation*}
The Stage 5 rules then halve the counter value:
\begin{align*}%
\{
& \dwrule{\osr{x}{{}}{5}}{\osr{x}{{}}{1}\esr{x}{{}}{1}},
\dwrule{\os{x}{{}}{5}}{\oes{x}{{}}{1}},
\dwrule{\osm{x}{{}}{5}}{\oesm{x}{{}}{1}},
\dwrule{\es{x}{{}}{5}}{\oes{x}{{}}{1}},
\dwrule{\esm{x}{{}}{5}}{\oesm{x}{{}}{1}},
\\
& \dwrule{\os{a}{{}}{5}}{\oes{a}{{}}{1}},
\dwrule{\es{a}{{}}{5}}{\oesm{a}{{}}{1}},
\dwrule{\thirdsymbol{a}{{}}{5}}{\epsilon},
\dwrule{\osm{a}{{}}{5}}{\oesm{a}{{}}{1}},
\dwrule{\esm{a}{{}}{5}}{\oesm{a}{{}}{1}}\}
\end{align*}
Continuing our computation we get:
\begin{equation*}%
\oesr{x}{0}{1}\;
\oes{x}{1}{1}\;
\oesm{x}{2}{1}
\ldots
\oesm{x}{l}{1}\;
\oes{a}{{}}{1}\;
\oesm{a}{{}}{1}\;
\oes{a}{{}}{1}\;
\oesm{a}{{}}{1} %
\ldots
\oes{a}{{}}{1}\;
\oesm{a}{{}}{1}
\end{equation*}
which switches control back to Stage 1.

Each iteration of the 5 stages halves the counter value. After $\log \Vert \hat{w},\oes{a}{{}}{{}} \Vert$ iterations the counter has value 1, this causes the output from Stage 4 to be of odd length for the first time. This in turn switches parity to even (dotted symbols) during Stage 5, which is detected at the beginning of Stage 1, by the rules:
\begin{equation*}%
\{
\dwrule{\esr{x}{{}}{1}}{\#},
\dwrule{\es{x}{{}}{1}}{\osr{x}{{}}{6}\esr{x}{{}}{6}},
\dwrule{\esm{x}{{}}{1}}{\oes{x}{{}}{6}},
\dwrule{\es{a}{{}}{1}}{\oes{a}{{}}{6}},
\dwrule{\esm{a}{{}}{1}}{\oes{a}{{}}{6}}
\}
\end{equation*}
The first of these rules deletes $x_0$ in one step. The second of these rules passes the bar forward by one symbol pair, while~2 of the others unmark the remaining symbols.
\begin{equation*}%
\oesr{x}{1}{6}\;
\oes{x}{2}{6}\;
\oes{x}{3}{6}
\ldots
\oes{x}{l}{6}\;
\oes{a}{{}}{6}\;
\oes{a}{{}}{6}
\ldots
\oes{a}{{}}{6}
\end{equation*}
The $\#$ symbol restores the parity to odd so that we read undotted symbols (in subsequent computations in this paper).

For our example iteration we chose $i=l$. If the dataword $\hat{w}$ is in the more general form $i \in\{0,\ldots,l\}$ given by the lemma statement then the same proof holds; our rules are such that embedding the counter inside the dataword does not affect parity in a way that would change the algorithm control flow.
\qed\end{proof}

The following lemma provides much of the mechanics required for simulation of the appending of a cyclic tag system appendant (point (i) from the introductory paragraph of Section~\ref{sec:sim}). Simulating the appending is straightforward, the main work is in maintaining the equality in Equation~\eqref{eq:counterValue}. Thus, after appending, if the simulated dataword has length that is at least the counter value then we double the counter to satisfy Equation~\eqref{eq:counterValue}. The Lemma is stated for the case that the counter value is initially the next power of two greater than the encoded dataword length.

\begin{lemma}\label{lem:simAppend}
Given a word of the form
\begin{equation*}%
\hat{w}=
\osr{1}{{}}{\dczero}
\esr{1}{{}}{\dczero}\;
\oes{x}{1}{\dczero}\ldots\oes{x}{{i}}{\dczero}\;
\oes{a}{{}}{\dczero} \;
\oes{a}{{}}{\dczero}
\ldots
\oes{a}{{}}{\dczero}\;
\oes{x}{{i+1}}{\dczero} %
\ldots\oes{x}{{l}}{\dczero}
\end{equation*}
where $\Vert\hat{w},\oes{a}{{}}{\,}\Vert=2^{\lceil\log (l+1)\rceil}$ and $i\in\{0,\ldots,l\}$, then there is a 2-tag system~$T$ that computes
\begin{equation*}%
\begin{aligned}
\hathatw = & \
\osr{1}{{}}{\dcend}
\esr{1}{{}}{\dcend}\;
\oes{x}{1}{\dcend}\ldots\oes{x}{{i}}{\dcend}\;
\oes{a}{{}}{\dcend}\;
\oes{a}{{}}{\dcend}
\ldots
\oes{a}{{}}{\dcend}\;
\oes{x}{{i+1}}{\dcend}
\ldots
\oes{x}{{l}}{\dcend}\; %
\oes{x}{{l+1}}{\dcend}
\ldots\oes{x}{{l+c}}{\dcend}
\end{aligned}
\end{equation*}
where~$c \leqslant 2^{\lceil\log (l+1)\rceil}$, $\Vert\hathatw,\oes{a}{{}}{\,}\Vert=2^{\lceil\log (l+c+1)\rceil}$. $T$ completes this computation in time~$O(l \log l)$.
\end{lemma}
\begin{proof}
By applying the rule
$$\{
\dwrule{\osr{1}{{}}{\dczero}}{\oes{x}{{l+1}}{\dczero} \ldots\oes{x}{{l+c}}{\dczero}\,\osr{1}{{}}{\dcone}\esr{1}{{}}{\dcone}}
\}$$ to $\hat{w}$ we get a word denoted $\hat{w}_0$ that is of a similar form to $\hathatw$ except that $\Vert\hat{w_0},\oes{a}{{}}{\,}\Vert=2^{\lceil\log (l+1)\rceil}$, i.e.\ the counter has not yet been updated to the correct value of $\Vert\hathatw,\oes{a}{{}}{\,}\Vert = 2^{\lceil\log (l+c+1)\rceil}$. The remainder of the proof is concerned with updating the counter.

We let~$\hat{w}_k$ denote the output of the $k^{\textrm{th}}$ iteration of the 4 stages. The rules for Stages $\dcone$ to $\dcfour$ are of a similar flavour to those used in the proof of Lemma~\ref{lem:passBar}, so we omit them in favour of a brief overview.

We begin by computing on $\hat{w}_0$. During Stages $\dcone$ and $\dctwo$ of iteration $k$, we compute $\Vert \hat{w}_k,\oes{a}{{}}{{}} \Vert = \Vert \hat{w}_{k-1},\oes{a}{{}}{{}} \Vert /2$,
by marking every second pair of $\oes{a}{{}}{{}}$ symbols (here we mark the even numbered pairs).  Then in Stages $\dcthree$ and $\dcfour$ we compute
\begin{equation}\label{eq:doubleCounterRecurrence}
\Vert \hat{w}_k,\oes{x}{{}}{{}} \Vert = \left\lfloor \frac{\Vert \hat{w}_{k-1},\oes{x}{{}}{{}} \Vert}{2} \right\rfloor
\end{equation}
by marking every second pair of $\oes{x}{{}}{{}}$ symbols (here we mark the odd numbered pairs). We then return to Stage $\dcone$ and iterate until $\Vert\hat{w}_k,\oes{a}{{}}{{}}\Vert = 1$:
the counter now has an odd value (for the first time) and we detect this in Stage~$\dcthree$. The number of fully completed iterations, and final value for~$k$, is~$k=\log \Vert \hat{w},\oes{a}{{}}{{}} \Vert = {\lceil \log (l+1) \rceil}$. An additional stage restores the parity (by introducing an extra~$\#$ symbol and then deleting it after one round) so that we are reading undotted symbols. Then $\hat{w}_k$ is of the form
\begin{equation*}
\begin{aligned}
\hat{w}_k = & \
\osr{1}{{}}{\dcfive}\esr{1}{{}}{\dcfive}\;
\oesm{x}{1}{\dcfive}
\ldots
\oesm{x}{{i}}{\dcfive}\;
\oes{a}{{}}{\dcfive}\;
\oesm{a}{{}}{\dcfive}%
\ldots
\oesm{a}{{}}{\dcfive}\;
\oesm{x}{{i+1}}{\dcfive}%
\ldots
\oesm{x}{{l}}{\dcfive}\;
\oesm{x}{{l+1}}{\dcfive}
\ldots
\oesm{x}{{l+c}}{\dcfive}
\end{aligned}
\end{equation*}

At this point the number $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert$ of unmarked $\oes{x}{{}}{{}}$ pairs satisfies $0\leqslant \Vert \hat{w_k},\oes{x}{{}}{{}} \Vert \leqslant 1$. To see this, note that $\Vert\hat{w}_0,\oes{a}{{}}{\,}\Vert=2^{\lceil\log (l+1)\rceil}$ and $c \leqslant 2^{\lceil\log (l+1)\rceil}$. Solving Equation~\eqref{eq:doubleCounterRecurrence} for $k=\lceil \log (l+1) \rceil$ gives~0 iff
$$0 \leqslant
\Vert \hat{w}_0,\oes{x}{{}}{{}} \Vert = l + c
< 2^{\lceil \log (l+1) \rceil}$$
and~1 iff
$$2^{\lceil \log (l+1) \rceil} \leqslant \Vert \hat{w}_0,\oes{x}{{}}{{}} \Vert  = l + c < 2^{\lceil \log (l+c+1) \rceil} $$%
There are no other possible values for $\Vert \hat{w}_0,\oes{x}{{}}{{}} \Vert$ thus we have only to check whether $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert$ is 0 or 1. To prepare, in two consecutive rounds we apply the rules
\begin{eqnarray*}
\{
\dwrule{\osr{1}{{}}{\dcfive}}{\osr{1}{{}}{\dcsix}\esr{1}{{}}{\dcsix}},
\dwrule{\os{x}{{}}{\dcfive}}{\threesymb{x}{{}}{\dcsix}},
\dwrule{\osm{x}{{}}{\dcfive}}{\oesm{x}{{}}{\dcsix}},
\dwrule{\os{a}{{}}{\dcfive}}{\oes{a}{{}}{\dcsix}},
\dwrule{\osm{a}{{}}{\dcfive}}{\oes{a}{{}}{\dcsix}}
\}
\end{eqnarray*}
and
\begin{align*} %
 \{
& \dwrule{\osr{1}{{}}{\dcsix}}{\osr{1}{{}}{\dcseven}\esr{1}{{}}{\dcseven}},
\dwrule{\os{x}{{}}{\dcsix}}{\oes{x}{{}}{\dcseven}},
\dwrule{\thirdsymbol{x}{{}}{\dcsix}}{\epsilon},
\dwrule{\osm{x}{{}}{\dcsix}}{\oes{x}{{}}{\dcseven}},
\dwrule{\esm{x}{{}}{\dcsix}}{\oes{x}{{}}{\dcseven}},\\
& \dwrule{\os{a}{{}}{\dcsix}}{\oes{a}{{}}{\dcseven}},
\dwrule{\es{a}{{}}{\dcsix}}{\oes{a}{{}}{\dcseven}},
\dwrule{\osm{a}{{}}{\dcsix}}{\oes{a}{{}}{\dcseven}},
\dwrule{\esm{a}{{}}{\dcsix}}{\oes{a}{{}}{\dcseven}}
\}
\end{align*}%
These rules have the effect of shifting the parity of the read head to dotted symbols iff $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert = 1$. In addition these rules unmark all marked symbols, as the marks are not needed below.

We have two cases:\\
Case 1: If $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert = 0$ we do not need to change the counter value in order to satisfy Equation~\eqref{eq:counterValue}. In this case we detect $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert = 0$ by reading the undotted, barred symbol $\osr{1}{{}}{\dcseven}$. To complete the computation we apply the rules
\begin{equation*}
\{
\dwrule{\osr{1}{{}}{\dcseven}}{\osr{1}{{}}{\dcend}\esr{1}{{}}{\dcend}},
\dwrule{\os{x}{{}}{\dcseven}}{\oes{x}{{}}{\dcend}},
\dwrule{\os{a}{{}}{\dcseven}}{\oes{a}{{}}{\dcend}}
\}
\end{equation*}
Case 2: If $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert = 1$ we double the counter to satisfy Equation~\eqref{eq:counterValue}. In this case we detect $\Vert \hat{w_k},\oes{x}{{}}{{}} \Vert = 1$ by reading the dotted, barred symbol~$\esr{1}{{}}{\dcseven}$. We then restore odd parity and double the counter value, by applying the rules
\begin{eqnarray*}
\{
\dwrule{\esr{1}{{}}{\dcseven}}{\#\osr{1}{{}}{\dcend}\esr{1}{{}}{\dcend}},
\dwrule{\es{x}{{}}{\dcseven}}{\oes{x}{{}}{\dcend}},
\dwrule{\es{a}{{}}{\dcseven}}{\oes{a}{{}}{\dcend}\;\oes{a}{{}}{\dcend}}\}
\end{eqnarray*}
\qed
\end{proof}

\subsection{Proof of main result}
\begin{theorem}\label{thm:2tagSimCycTag}
Given a cyclic tag system~$C$ that computes in time $t(n)$ on input of length~$n$, where~$n$ is at least the length of~$C$'s longest appendant, then there is a 2-tag system~$T_C$ that simulates the computation of~$C$ and computes in time~$O(t^{2}(n)\log t(n))$.
\end{theorem}
\begin{proof}
As stated, it is assumed that~$C$'s input length $n$ is at least that of its longest appendant. As part of the 2-tag input encoding in Definition~\ref{def:2tagInputEnc}, the constant number of shorter inputs are assumed to be padded to this length.

Recall, from the beginning of Section~\ref{sec:sim}, the decomposition of a single cyclic tag computation step into the conceptual steps~(i),~(ii) and~(iii).

Lemma~\ref{lem:simAppend} provides the algorithm for step (i) for the case that $\osr{x}{0}{{}}=\osr{1}{{}}{{}}$. For the other case of $\osr{x}{0}{{}}=\osr{0}{{}}{{}}$ we skip (i). Deciding between the two cases is easily implemented by setting the parity to even iff $\osr{x}{0}{{}}=\osr{1}{{}}{{}}$.

Lemma~\ref{lem:passBar} provides the main mechanics for step (ii). After applying Lemma~\ref{lem:passBar} the barred pair $\osr{x}{{}}{{}}\esr{x}{{}}{{}}$ is either on the left of the counter ($i>0$ in the lemma statement), or else has jumped over the counter and is now on the right ($i=0$).
Lemma~\ref{lem:simAppend} assumes the head is on the left. Therefore if the head is on the right we want to move it to the left so that we can apply Lemma~\ref{lem:simAppend} in subsequent iterations. Therefore as a second part of step (ii) we do the following. In one round we read the entire dataword~$\hat{w}$, and reproduce~$\hat{w}$ exactly except that the underindex is incremented and the length of the barred subword is increased from~2 to~3 by a rule of the form
$\dwrule{\osr{x}{{}}{{}}}{\osr{x}{{}}{{}}\esr{x}{{}}{{}}\thirdsr{x}{{}}{{}}}$. On the one hand, if in the next round we read dotted counter symbols ($\es{a}{{}}{{}}$) then the head is to the left of the counter; then in one more round we set the parity to odd and we're done. On the other hand, if we read undotted counter symbols ($\os{a}{{}}{{}}$) then we set the parity to odd in one round and furthermore we read the counter an extra time to place it to the right of the head.

Lemata~\ref{lem:passBar} and~\ref{lem:simAppend} state that the counter value should be the next power of two greater than $C$'s dataword length $l$. It is not difficult to check that their proofs also hold for the more general case where the counter value is any power of two greater than  $l$. Therefore these proofs are still applicable if $C$'s dataword drastically decreases in length. In this case the big-Oh time bounds in these proofs depend on the counter value rather than $l$. But since the counter value is~$O(t(n))$ this does not affect our overall complexity analysis.

For step (iii) we introduce a new decoration for 2-tag symbols.
So far, the number $q$ of distinct 2-tag symbols that we have used is dependent on our algorithm. We now increase this number to $pq$ where, as usual, $p$ is the number of appendants of $C$. We create a new symbol set by decorating each 2-tag symbol
$y \in \{
\osr{x}{{}}{{}},
\esr{x}{{}}{{}},
\os{x}{{}}{{}},
\es{x}{{}}{{}},
\thirdsymbol{x}{{}}{{}},
\osm{x}{{}}{{}},
\esm{x}{{}}{{}},
\os{a}{{}}{{}},
\es{a}{{}}{{}},
\thirdsymbol{a}{{}}{{}},
\osm{a}{{}}{{}},
\esm{a}{{}}{{}},
\#
\}$ with an integer ${\arbApp}$ for all $0 \leqslant {\arbApp} < p$. Using this, our encoding of an arbitrary cyclic tag system configuration is of the form
\begin{equation*}%
\osra{x}{0}{j}{{\arbApp}}
\esra{x}{0}{j}{{\arbApp}}\;
\osa{x}{1}{j}{{\arbApp}}
\esa{x}{1}{j}{{\arbApp}}
\ldots
\osa{x}{{i}}{j}{{\arbApp}}
\esa{x}{{i}}{j}{{\arbApp}}\;
\osa{a}{{}}{j}{{\arbApp}}
\esa{a}{{}}{j}{{\arbApp}}\;
\osa{a}{{}}{j}{{\arbApp}}
\esa{a}{{}}{j}{{\arbApp}}
\ldots
\osa{a}{{}}{j}{{\arbApp}}
\esa{a}{{}}{j}{{\arbApp}}\;
\osa{x}{{i+1}}{j}{{\arbApp}}
\esa{x}{{i+1}}{j}{{\arbApp}} %
\ldots
\osa{x}{{l}}{j}{{\arbApp}}
\esa{x}{{l}}{j}{{\arbApp}}
\vspace{1.5ex}\end{equation*}

Steps (i) and (ii) are simulated, while ignoring the value of ${\arbApp}$. (Note that our 2-tag algorithms are easily concatenated by having appropriate~$j$ values at the beginning and end of each algorithm.) Then~$j$ is given a value that signals the completion of steps~(i) and~(ii). Then step~(iii) (incrementing the program marker) is simulated by rules of the form
\begin{equation*}
\{
\dwrule{\osra{x}{{}}{j}{\arbApp}}{\osra{x}{{}}{j}{{{\arbApp}'}}\,\esra{x}{{}}{j}{{{\arbApp}'}}},\,
\dwrule{\osa{x}{{}}{j}{\arbApp}}{\osa{x}{{}}{j}{{\arbApp}'}\,\esa{x}{{}}{j}{{\arbApp}'}},\,
\dwrule{\osa{a}{{}}{j}{\arbApp}}{\osa{a}{{}}{j}{{\arbApp}'}\,\esa{a}{{}}{j}{{\arbApp}'}}
\}
\vspace{1.5ex}\end{equation*}
where ${\arbApp}'=({\arbApp}+1)\mod p$.
Applying these rules for a given $k$ takes only one round, or $O(l)$ timesteps.

In the time analysis of the computation of $T_C$ note that for an arbitrary timestep of $C$ we have $l=O(t(n))$. Therefore via Lemmas~\ref{lem:simAppend} and~\ref{lem:passBar}, and the present proof, $T_C$ simulates a {\em single} step of $C$'s computation in time\linebreak $O(t(n) \log t(n))$.\qed
\end{proof}

We get the proof of Theorem~\ref{thm:2-tagSimTM}, our main result, by combining the statements of Theorems~\ref{lem:CycTagSysSimTM} and~\ref{thm:2tagSimCycTag}. If we combine these Theorems directly we get a time bound that is higher than that of Theorem~\ref{thm:2-tagSimTM}. To get our tighter bound a more careful analysis is required where we substitute space bounds for time bounds whenever possible in the analysis.

\section*{Acknowledgements}
Many thanks to Matthew Cook for finding an omission in the proof of Theorem~\ref{thm:2tagSimCycTag}. Both authors are funded by the Irish Research Council for Science, Engineering and Technology.

redpajama_set_name: RedPajamaArXiv


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