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Well received in good order. Thank you.
Well received from a friendly and responsive seller! Highly recommended!
A painless purchase. Items arrived safely from US on 4th Sept in Singapore for an order placed on Sun 26th Aug.
Excellent Service delivery as assured and product in great condition and all original items as mentioned in the description.
Good communication about impending shipping and prompt delivery from the US. Very happy with the purchase.
How many pcs with the pack 1 and pack 2?
1 pack means 6 pcs.


It's no secret that I'm a true loyalist of the Club ITC Culinaire membership of the ITC Hotels. For business savings or personal spends, the membership's got your back.
My recent saving with the card was during an office party I hosted. My boss had decided to start a new innings in her career after completing 20 glorious years with our company. It was only befitting to host a dinner honouring her contribution to the company. I invited a group of 25 of her closest associates for cocktails and dinner at K&K at the ITC Grand Central in Mumbai.
At K&K, show kitchens visible from all corners of the dining area are the highlights where the sights and smells of cooking easily entice your senses. Our farewell bash was a sit down dinner with a pre-set menu consisting delighful kebabs, curries and biryanis. Saboot Tandoori Aloo, Dal Bukhara and Tarkari Pulao were the winners with our group.
I foot a bill of Rs. 1,80,000 for a gathering of 25 people. Club ITC Culinaire got me 45,000 Points which amounts for 25% of the total bill amount. These points are enough to cover my cost for the next 5 visits to the hotel. I might even sneak in a room night stay with my family over a long weekend at 6,000 Points for a room night stay.
It's a rewarding feeling to think about your savings from one dinner covering the cost of your next 5 visits.
I constantly jet set around the country and the region for work. Even though business travel is just a part of the grind for most people, it is something I'm extremely meticulous about.
My travels over the past few years have taught me that choosing the right loyalty card is a crucial task on your way to start reaping the rewards from extensive dining and stays at hotels. For anybody who stays at hotels for extended periods of time, it only makes sense to get rewarded for the same.
I chose the Taj Hotels as it provides me access to a collection of exquisite properties across the region. My normal stops are mostly large Indian cities like Delhi and Mumbai. There is a Taj property in each of these places and they really look after their guests.
The Epicure membership comes with great perks like a complimentary night stay (which I use for my holiday) and point earning and redemption over food, drinks and the spa. Added to the 25% Epicure point earning on food, the membership offers Happy Hours at the Bars all days of the week. When I travel alone, I do tend to use the Happy Hours offer often. The best bit is that I meet other single travellers and many use the Epicure membership for the Happy Hours too. It's a super benefit.
I accumulate enough Epicure Points and Airline miles due to my incessant travel that I can treat my family to two holidays a year at almost no cost. The Epicure Points take care of the stay and the earned miles take care of our travel.
The Epicure membership comes with great perks like a complimentary night stay (which I use for my holiday) and point earning and redemption over food, drinks and the spa.
The annual membership price is recovered by the use of the free room night and the 50% Point earning for the 1st three transactions alone.
I'd recommend the membership to anyone with a penchant for travel and entertainment. It works well to reward your family and make up for the time spent away with some wonderful memories at the Taj, all earned through redemption of Points. The Taj has a wonderful set of experiences you can redeem your Points for. If you're interested to know more, you can log on to the Taj website for a detailed view on their Point earning and redemption system.
I'm a 40 year old professional working as a Partner with a Big Four accounting firm and I'm a huge advocate of networking through business entertainment. This is not just a business need but also a great way to build a relationship with clients or prospective clients.
With my fair share of experience in entertaining clients for business, I have slowly discovered that points based loyalty cards are just perfect. One earns points each time you entertain someone just like you earn miles when you fly on business. Once I have enough points, I redeem them to take my family out for a meal at no cost or for a quick vacation paid for by my points.
The best part is the 6000 preloaded points that come with the card when buying or renewing the membership.
Off all the dining options available at ITC Hotels, their Indian food is my weakness, hence I'm a regular at Bukhara and Dum Pukht. I earlier had the ITC discount based membership and was very disappointed when they switched to the points program. Due to my love for the food, I got myself a Club ITC Culinaire membership anyway and discovered the wonderful benefits of the points based programs. I earn points while dining and also when I stay at the ITC hotels during my travels across the country. You can't even believe how quickly these points add up.
The best part is the 6000 preloaded points that come with the card when buying or renewing the membership. My wife and I go to celebrate by redeeming these points for a one night stay at ITC Maurya at no cost, virtually recovering my investment in the annual fee and the night stay is only the beginning of the point earning journey.
Let me tell you an incident in particular to explain how wonderful this membership is. Last year I spent about Rs. 80,000 on dining and entertainment and about Rs. 1 lakh on my stay bills for which I earned a total of 25,000 Green Points, 25% on dining (20,000 points) and 5,000 points on stay. I used these points for a 2-night getaway with my family to Agra at the wonderful ITC Mughal and 3 nights at ITC Rajputana, the charming ITC hotel in Jaipur, using up 5000 points per night.
Folks, there are some more benefits like special room rates, discounts on welcome break, birthday bonuses and more. It works all round. Go enroll and enjoy the point earning opportunity.
Connoisseur Club for connoisseurs of a good lifestyle!
The Leela has a range of award-winning properties that celebrate the country's diverse geography and rich architectural history. Their luxury palaces, hotels and resorts are present in Mumbai, Bengaluru, Kovalam, Gurugram, Delhi and India's magical holiday destinations of Goa and Udaipur. In many ways, The Leela has redefined gourmet food, luxurious stays and indulgent living.
Hand crafted menus across restaurants at Leela hotels spell culinary excellence. With a selection of restaurants like Jamavar, The Great Wall, Citrus, Spectra and others, The Leela offers the best in dining. While Jamavar brings royal culinary traditions to the city, Spectra and Citrus bring to you the best of global cuisine.
Connoisseur Club offers undeniably good discounts on food and drinks, up to a 50% off on dining and a 15% on beverages across the hotels.
My company got for me and some of my colleagues a Connoisseur Club membership as we were regular visitors to the corporate hubs of Delhi, Mumbai and Bengaluru. The company felt that using the membership would lead to corporate savings on dining and living and indeed it did!
Connoisseur Club offers undeniably good discounts on food and drinks, up to a 50% off on dining and a 15% on beverages across the hotels in these three business hubs. More so, each time I booked my hotel room, I got a discount on the Best Available Rate on accommodation. Group dining certificates and others added to a reduction in our business expense as well.
The best part was that the membership came with a free room night stay in my city, which of course I could use as a personal gift from the company. I took advantage of the same to go stay at The Leela in Gurugram with my family. During our stay we conveniently shopped at the adjoining Ambience Mall and later dined at some of the finest restaurants in Gurugram at The Leela. Skimming through the vouchers made me notice the discounts offered on lifestyle services and got me to try the Spa which was truly rejuvenating. This membership really indulges you to experience the magnificence of the Leela hotels. The membership also comes with a spouse card at no extra charge.
If you're a fan of beautiful locations and world class service at a reasonable value, go ahead and get this membership or try your luck and nudge your company to get one for you.
I have always wanted to pamper my parents, how they pampered us, if not always, at least on the big day, their Anniversary!! I never felt satisfied with what I've gifted them because it's no different from what I do usually, as the options are limited to conventional gifts like flowers, cards or dinners.
I was looking for the perfect gift which would truly surprise and overwhelm my parents and that's when my friend suggested the Connoisseur Club membership from The Leela Palace Bengaluru where she's a member. She recommended Connoisseur Club by saying that "It's a complete package worth every penny".
I quickly researched the Connoisseur Club membership and I have to say it was an ideal gift for my parents and they could do all they ever wanted with the membership. The Platinum Level membership caught my attention in particular, I felt like it was crafted just for my needs. Off all the variants the membership offered, Platinum Level was the most indulgent and high value membership.
The membership kit arrived at my doorstep within a few days, ready to be gifted to my parents. My membership was so well packaged that it exuded elegance and class, it was perfect to be gifted as it is.
It's a delight to hear my parents talk about The Leela Palace Bengaluru and their savings courtesy Connoisseur Club. My parents' most recent experience with Connoisseur Club is their two night stay in a suite at The Leela Palace Bengaluru. They were immensely impressed with the hotel façade drawing inspiration from the architectural style of the Royal Palace of Mysore, and exclusive access to the Club Lounge, which enabled them to enjoy multiple rounds of drinks and snacks at the Club Lounge and exploit their taste for whiskey sour each evening they stayed at the hotel.
On their second day at the hotel, we invited friends and family to surprise them, we were a total of 8 people on a table at Zen, the Pan Asian restaurant and guess what, just because my parent's were now members we got a 50% discount over a sumptuous brunch with alcohol. A discount of this capacity was almost unheard of to most of us on that table!
My parents were very pleased and said this was one of the best experiences of their life. I totally agree with them, they did all that they wanted to do immersed in royalty and luxury while saving big bucks. I'm sure that they've already managed to save INR 30,000 or more due to this membership.
Since it's a year long membership, we look forward to visit The Leela Palace Bengaluru again and again to experience it all! This is just the beginning; I would love to try the other Leela properties too.
Club ITC Culinaire for the gourmet aficionado!
I was on a hunt for the perfect birthday present for my husband when my friend suggested the Club ITC Culinaire membership to me, keeping in mind his absolutely irrevocable love for food and all things good in life.
It was his big 50th, I knew the gift had to be special, but a hotel membership is something I was really unsure of only because I had never heard of anything like it before. However, one look at their online brochure was enough for me to make a decision and buy the membership.
ITC has the most beautiful properties across the country. From ITC Maurya nestled in greenery in the heart of Delhi to the very royal ITC Rajputana in Jaipur, each property is worth a visit for the unique experience it has to offer. In addition to beautiful hotels, ITC Hotels are known to create handcrafted signature dining experiences at their iconic restaurants like Bukhara, Dum Pukht, Peshawari and others. The restaurants known for their flavorful excellence offer the finest in food, service, ambience and culinary traditions.
Club ITC Culinaire membership Green Points can be accumulated and redeemed on accommodation and lifestyle at over 100 hotels in 75 locations and 80 Wills Lifestyle stores.
My husband absolutely loved the membership card and soon enough I knew that I had made the right choice. The welcome bonus of 6000 Green Points allowed us to redeem them for a one night stay at ITC Maurya to celebrate my husband's 50th.
The best part of the evening was that in addition to the 25% Green Points, we earned 50% bonus points on the entire bill as a part of the birthday week redemption and since our total expenditure crossed 25,000, we were credited with another 1,000 Points. On a bill of Rs 30,000 we had earned 12,250 points, which can now be used for 2 room night stays at ITC Maurya itself at 6000 points a room night.
We had invited a few friends for dinner at Bukhara to celebrate the special occasion and enjoyed a very memorable meal as the restaurant never fails to impress. The best part of the evening was that in addition to the 25% Green Points, we earned 50% bonus points on the entire bill as a part of the birthday week redemption and since our total expenditure crossed 25,000, we were credited with another 1,000 Points. On a bill of Rs 30,000 we had earned 12,250 points, which can now be used for 2 room night stays at ITC Maurya itself at 6000 points a room night or at another similar ITC Hotel across India or a visit back to Bukhara for knocking off Rs 12,250 from our next bill.
Even though the membership was a gift for my husband, the complimentary spouse card lets me enjoy the benefits too! Now I use my spouse card to redeem points every time I have a Kitty to attend or go out for a casual lunch or evening with my friends.
Club ITC Culinaire is a combination of a good deal and super experiences at an ITC Hotel. From 25% Green Points on dining, birthday week celebration bonus of 50% points (including the spouse birthday), to 5% earnings on your stay and on expenditure at Wills Lifestyle stores, the membership caters to all aspects of a distinguished lifestyle. Perfect for connoisseurs of good food, fashion and experiences!
Last month we had my brother and his family visit us from Mumbai. As they landed in Kolkata, we told them about the relaxing staycation we had planned for all of us at the Radisson Kolkata Ballygunge courtesy the Rezerved membership.
The day we were to check in, everyone woke up excited with their plans ready for the weekend. The ladies had plans to hit the spa while the men planned to spend quality time with the kids within the indulgent abode that was our hotel.
Radisson Kolkata Ballygunge is one of the most premium hotels in Kolkata with the best rooms in the city and good in-house food and beverage options. The hotel houses some of the best food options at the all day diner, K 19 and the cozy Smoke Shack which serves some very creative kebabs.
We had booked two rooms since we were 2 families of 3 members each. Thanks to the Rezerved membership we got the first room complimentary and the second room at a special price of only Rs. 3000.
The Rezerved benefits didn't just end at accommodation. We were also able to redeem our meal certificates for a 50% discount on the food bill for a table of six over two meals during our stay. Not to forget, the women also got 20% discount at the Spa for their much awaited spa sessions.
With young children it was inevitable to avoid room service every couple of hours to curb their hunger pangs. Luckily for us, Rezerved card discounts were applicable during all meal periods.
Thanks to the Rezerved membership we got the first room complimentary and the second room at a special price of only Rs. 3000.
Even though we exhausted most of our certificates, we plan to visit the hotel often enough to enjoy discounted meals and food promotions with the help of our card benefits. The card discounts offer up to a 50% discount at the restaurants depending on the number of guests.
Rezerved managed to make our relaxing staycation absolutely memorable. We had the most gala time at almost half the cost of the original services. My brother was so impressed with the membership and its benefits that he plans to buys the same for himself.
Retail therapy is the real deal for me and my standard stop is the Ambience Mall in the heart of Gurugram. On one of my shopping trips, I stepped into Spectra at The Leela Ambience Gurugram Hotel for lunch with a friend. We thoroughly enjoyed the extravagant buffet with a choice of cuisine for each. On my way out, I crossed paths with a Leela executive who told me about the Connoisseur Club membership. It seemed so attractive that I got it immediately and have never regretted it ever since. Now it is the most important card in my wallet, almost as important as my credit cards.
The most attractive bit of the membership are the suite night stays built into the Platinum Level. You can use these one at a time and I have used up all the three certificates, enjoying the super luxury of a suite at the hotel, dining discounts at Spectra and the walk straight into the Ambience Mall. For the ones who don't know, the hotel and the Mall are a part of the same complex. In addition to the dining and accommodation certificates, I ended up using the spa benefit of the membership at the hotel as well.
Now it is the most important card in my wallet, almost as important as my credit cards.
The membership has more to offer at the lovely Leela hotels in Udaipur, Goa or Kovalam and if you want a city experience then of course the stunning Leela Palace hotels at Bangalore and Chennai. What's even better is that the card works for dining discounts at the Leela hotel in East Delhi as well, a place I end up going to every couple of months. Highly recommended if you wish to lift your lifestyle with The Leela Hotels.


text:
Svatoslav est un toponyme qui peut désigner :

Svatoslav, une commune du district de Brno-Campagne, dans la région de Moravie-du-Sud ;
 Svatoslav, une commune du district de Třebíč, dans la région de Vysočina.

Homonymie de localités en  Tchéquie

redpajama_set_name: RedPajamaWikipedia


-Self contained certificate : Possibility to do Freedom Camping!
Campervan very comfortable and convenient to live and travel in couple or alone. All the accessories are less than 1 year old. Curtain linings cracking a bit.
- Cutlery, plates (x4), glasses (x3), cups (x4), saucepans (x2) pans (x2), dish.
Nice chunky and wrinkly puppies. Very well socialized. All Vet checked, micro chipped and wormed to date. Excellent colors and markings. contact for details if interested.
With the best promotional staffs and event staffs, you can easily take off your brand popularity. They will help in creating fans for your brand that will easily spread positive reviews about your brand in the community.
Shining vinyl and chips floor with UV coating is our speciality, the epoxy flooring in Brisbane is served by us, commercial or home garage flooring we are perfect for every type of epoxy flooring.


text:
Aguon renews effort to repeal pay raises
(Press Release) The effort to repeal the raises of Guam's top elected officials moves forward in the 34th Guam Legislature with Senator Frank B. Aguon, Jr.'s introduction of Bill No. 4-34. If successful, the new legislation will revert the annual salaries of the governor, lieutenant governor and senators to annual salary amounts collected prior to the enactment of Public Law 32-208 passed in November, 2014.
"We've seen several versions of this raise repeal fail in the past two years because of one reason or another. I think it's high time that the island's top elected officials set the example. While a full repeal of 32-208 is preferred, we just keep hitting a brick wall. If we want to effect change - let's start with ourselves, from the top down and start chipping away at this brick wall," Senator Aguon said. "Bill No. 4-34 will affect only the governor, lieutenant governor and senators. Additional measures can be taken to reduce the salaries of other elected and politically-appointed officials with separate legislation. But let's start here - with us, right now."
Bill No. 4-34 seeks to exclude I Maga'låhen Guåhan, I Segundu na Maga'låhen Guåhan and senators of I Liheslaturan Guåhan from the Competitive Wage Act of 2014, setting the salaries of the governor at a rate of Ninety Thousand Dollars ($90,000) per annum, the lt. governor at a rate of Eighty-Five Thousand Dollars ($85,000) per annum, and senators of the Guam Legislature at a rate of Fifty-Five Thousand Three Hundred Three Dollars ($55,303) per annum. Unified Pay Schedule/Executive Pay Schedule for the governor, lt. governor, and senators was authorized by the Civil Service Commission on October 3, 1991[revised on April 30, 1996].
"If you take a look at news stories from our local media outlets online and if you scroll through the comments on social media sites, you'll see one resounding theme no matter what the issue is: the people of Guam want these raises repealed,"
Senator Aguon said. "Tax refund payments are on the horizon, the court's decision to affirm the $14 Million award to Guam YTK, Simon Sanchez High School needs to be rebuilt and the people feel we haven't put these issues first. Let's repeal these raises quickly and move forward."
Bill No. 4-34 was introduced this morning. Per the 34th Guam Legislature's Standing Rules, the Bill must be referred by the Committee on Rules to the "Principal Committee" prior to any public hearing. The Principal Committee must convene a public hearing for Bill No. 4-34 prior to the bill being placed on the legislature's upcoming session agenda. As of today, no date has been set for the legislature's next session.
Guam 34th Legislature
Guam News & Features
Guam is the second most expensive place to buy eggs in the US
Moylan: VA must cover shipping costs of adaptive vehicles for Guam veterans
Once again, Guam governor asked to report emergency procurements

redpajama_set_name: RedPajamaCommonCrawl


ATLANTA, GA--(Marketwire - February 1, 2011) - The 2011 Southeast Venture Conference (SEVC) announced today, the second round of companies selected to present at the March 2-3rd event being at the Ritz Carlton Buckhead in Atlanta, Georgia.
SEVC 2011 showcase companies will present to a sold out audience of venture capital investors, entrepreneurs, private equity professionals, investment bankers and others from the high growth technology and venture capital community.
A third and final round of companies will be announced prior to the conference, which will highlight over 50 showcase presenters from the Southeast and Mid-Atlantic regions.
In addition to showcase company presentations, the conference will feature a number of market discussion and best practice oriented panels, featured speakers and extensive networking opportunities.
Panels will discuss such topics as the forecast for the Venture Capital investment market, the IPO and alternative secondary markets, early stage startup fundraising strategies, raising a venture fund in the current environment and M&A strategies among others.


text:
Boeing 737 MAX to stay grounded till August
epa07484135 (FILE) - Pieces of the wreckage of an Ethiopia Airlines Boeing 737 Max 8 aircraft are piled at the crash site near Bishoftu, Ethiopia, 13 March 2019. Ethiopian Airlines flight ET 302 carrying 149 passengers and 8 crew was en route to Nairobi, Kenya, when it crashed on 10 March 2019. All passengers and crew aboard died in the crash. Media reports on 04 April 2019 state that the preliminary report into the crash of an Ethiopian Airlines plane ET 302 last month says the aircraft nose dived several times before it crashed. Ethiopian Airlines state the preliminary report clearly showed that the Ethiopian Airlines Pilots who were commanding Fligh tET 302 /on 10 March have followed the Boeing recommended and FAA approved emergency procedures to handle the most difficult emergency situation created on the airplane. Despite their hard work and full compliance with the emergency procedures, it was very unfortunate that they could not recover the airplane from the persistence of nose diving. As the investigation continues with more detailed analysis, as usual we will continue with our full cooperation with the investigation team. EPA-EFE/STRINGER
The International Air Transport Association (IATA) expects it could take until August before the Boeing Co 737 MAX returns to service, the airline group's head said on Wednesday, adding that the final say on the timing rested with regulators.
The boss of the airline industry's largest trade group Wednesday said the Boeing 737 MAX fleet could remain grounded for another two months or more in the wake of two crashes.
The airline industry has had a tough six months with fuel, labour and infrastructure costs increasing and trade tensions rising in addition to the 737 MAX grounding, de Juniac said.
At an IATA meeting for 737 MAX operators in Montreal last week, airline members said they wanted regulators to cooperate closely on the decision for the plane's re-entry to service.
Via Reuters / WSJ
Tags: Boeing, IATA
Previous House of Commons Speaker Bercow pledges to stay on
Next Di Maio calling for vote of confidence after European elections results in Italy


text:
human engagement counters misinformation
Posted 2021-03-15 ; filed under Complexity, Democracy.
A recent study conducted by Facebook suggests that when it comes to vaccine doubts and misinformation, "a small group appears to play a big role in pushing the skepticism".
Some of the early findings are notable: Just 10 out of the 638 population segments contained 50 percent of all vaccine hesitancy content on the platform. And in the population segment with the most vaccine hesitancy, just 111 users contributed half of all vaccine hesitant content. —WaPo 2021-03-14
Small groups of people can have influence beyond their numbers. For example when a committed minority in society rises above 25% there can be a tipping point. However it only takes 10% if those people have an unshakeable belief in their cause. Meanwhile, inside an organization, there is usually a small group of people — 3% — who can influence up to 85% its members. Find out more at — 25-10-3.
Too often those in power try to counter misinformation with facts. This does not work. A November 2019 article in the British Medical Journal by Kathryn Perera et al showed how difficult it is to change peoples' minds with regards to vaccinations. Facts don't change peoples' minds.
Lesson 2: don't bring a fact to a narrative fight
Experts and health professionals can arm themselves with white papers, peer reviewed studies, and symposia; but if these are our only weapons, we will only ever get so far. In an era in which experts are increasingly distrusted, the "we know best" mindset is counterproductive.
Those wishing to encourage vaccination need to identify and amplify the stories that emerge from the real lives and lived experiences of people in their communities (to start, they need to listen for them). It is no coincidence that the most effective climate advocacy in the world right now comes from the improvisations and stories of a 16 year old girl rather than the strategic plans of a generations old institution. —BMJ: New Power versus Old
A program in Québec, Canada shows what works — first connecting as a fellow human. EMMIE (l'Entretien Motivationnel en Maternité pour l'Immunisation des Enfants), in English — Motivational Interviewing in Maternity Hospitals for Child Immunization.
It [EMMIE] uses a method of counselling called motivational interviewing, an empathic, non-confrontational communication style developed by psychologists to bring about behavioural change.
"We are here to understand them and to give them the answers they need to make a good decision about immunization, and to help them to build a stronger decisional process about immunization of the child," said [Doctor] Gagneur … [In the case of one mother] Just 19 and living in rural Alberta at the time she had her first child, Coulis said she did a lot of reading, a lot of which she didn't understand fully. "I felt confused and unable to parse the information about this study versus that study." There were big claims by anti-vaccination literature, but Coulis didn't see those acknowledged by the government materials or by public health nurses … "The message from public health was, 'Just trust us, we've done the thinking for you,'" she said. —CBC 2020-12-19
There is a major disconnect between those who run our institutions and the realities of the 99%. Does the following story remind you of anyone in a leadership position? If so, then that is sad.
I was working with the head of nursing who was also responsible for performance and training for all nursing staff in a large hospital. I asked to do an on-site performance analysis over several days. As a visitor I had to be accompanied by a member of the hospital staff. This senior clinician took me around to all floors and let me interview a number of staff. What was surprising to me was learning that after two years on the job, it was the clinician's first time on the wards they were responsible for — in the same hospital!
This pandemic has shown the abject failure in communications from many of our institutions and authorities around the world. First they have to get out of their 'ivory towers' and meet people where they are. Leadership by walking around — physically or virtually —is necessary in our connected world. If our experts are not helping society make better decisions about health care, then what are they doing?
Source: Damian Corbet quoting Janice Kaffer CEO of Hôtel Dieu Grace Healthcare, Windsor, ON
the knowledge artisan
From Classroom to Boardroom


text:
Ava oder Awa ist ein weiblicher Vorname unterschiedlicher Herkunft.

Herkunft und Bedeutung
Der Name Ava hat verschiedene Herleitungen.

Im englischsprachigen Raum wird Ava [] als Variante von Eva angesehen.

Der persische Name Ava آوا [] bedeutet "Stimme" oder "Klang".

Des Weiteren kann es sich bei Ava [] um die Kurzform verschiedener germanischer Namen gesehen werden, die mit dem Element avi beginnen. Die Bedeutung von avi lässt sich nicht bestimmen, möglicherweise bedeutet es "begehrt", "gewünscht".

Möglicherweise geht der Name auch auf altsächsische Wort aval "Kraft", "Stärke" zurück.

Eine weitere Herleitungsmöglichkeit hält Ava für eine lateinisierte Form von Awe. Dieser Name geht wiederum aufs altgermanische Element *awja "Hagel" zurück.

Außerdem kann man Ava als eine Kurzform aller mit -ava endenden Namen sehen.

Darüber hinaus existiert Ava in Grönland auch als männlicher Vorname, dort mit der Bedeutung "kleiner Küstengeist".

Verbreitung
In den letzten Jahren stieg die Beliebtheit des Namens Ava in Deutschland kontinuierlich an. Im Jahr 2021 belegte Ava Rang 106 der beliebtesten Mädchennamen in Deutschland.

Namenstag
 7. Februar nach der ersten namentlich bekannten deutschsprachigen Dichterin Frau Ava (Ava von Melk / Ava von Göttweig, Niederösterreich).
 29. April
 6. Juni

Namensträgerinnen

 Ava (Dichterin) (* um 1060; † 1127), erste namentlich bekannte Dichterin deutscher Sprache, zuletzt als Inkluse in Österreich
 Ava Addams (* 1979), französische Pornodarstellerin
 Ava Willing Astor (1868–1958), US-amerikanische High Society-Lady
 Ava Celik (* 1990), deutsche Schauspielerin
 Ava Dalush (* 1989), britische Pornodarstellerin
 Ava DuVernay (* 1972), US-amerikanische Filmregisseurin
 Ava Gardner (1922–1990), US-amerikanische Schauspielerin
 Ava June (1931–2013), britische Sopranistin
 Ava Max (* 1994), US-amerikanische Popsängerin und Songwriterin
 Ava Mendoza (* 1983), US-amerikanische Jazzgitarristin
 Ava Ohlgren (* 1988), US-amerikanische Schwimmerin
 Auður Ava Ólafsdóttir (* 1958), isländische Schriftstellerin
 Ava Reed (* 1987), deutsche Schriftstellerin
 Ava Sophie Richter (* 2006), deutsche Kinderdarstellerin
 Ava Rose (* 1986), US-amerikanische Pornodarstellerin
 Ava Sunshine (* 2002), US-amerikanische Skirennläuferin
 Ava Vincent (* 1975), US-amerikanische Pornodarstellerin

Einzelnachweise

Weiblicher Vorname
Persischer Personenname


text:
\section{Introduction}

Belief change and belief merging have been topics of interest in
Artificial Intelligence for three decades
\cite{AGM85,KM91,KP02}. However, the restriction of such operators to
specific fragments of propositional logic has received increasing
attention only in the last years
\cite{DelgrandeSTW13,CreignouPPW14,CreignouPRW14,ZhuangP12,ZhuangPZ13,ZhuangP14,DelgrandeP15,HaretRW15}. Mostly,
the question tackled in these works is ``How should rationality
postulates and change operators be adapted to ensure that the result
of belief change belongs to a given fragment?''.  Surprisingly, the
question concerning the extent to which the result of a belief change
operation can deviate from the fragment under consideration has been
neglected so far.  In order to tackle this question, we focus here on
a certain form of reverse merging.  The question is, given an
\emph{arbitrary} knowledge base $K$ and some IC-merging (i.e.\ merging
with integrity constraint, see \cite{KP02}), operator $\Delta$ can we
find a profile $E$, {\em i.e.} a tuple of knowledge bases, and a
constraint $\mu$, both from a given \emph{fragment} of classical
logic, such that $\Delta_\mu(E)$ yields a result equivalent to $K$? In
other words, we are interested in seeing if $K$ can be distributed
into knowledge bases of simpler structure, such that the task of
merging allows for a reconstruction of the original knowledge. We call
this operation {\em knowledge distribution}.

Studying the concept of knowledge distribution can be motivated from
different points of view.  First, consider a scenario where the
storage devices have limited expressibility, for instance, databases
or logic programs. Our analysis will show which merging operators are
required to reconstruct arbitrary knowledge stored in such a set of
limited devices.  Second, distribution can also be understood as a
tool to hide information; only users who know the used merging
operator (which thus acts as an encryption key) are able to faithfully
retrieve the distributed knowledge.  Given the high complexity of
belief change (even for revision in ``simple'' fragments like $\mathit{Horn}$
and $\mathit{2CNF}$ \cite{EG92,LS01,CreignouPW13}), brute-force attack to
guess the merging operator is unthinkable. Finally, from the
theoretical perspective our results shed light on the power of
different merging operators when applied to profiles from certain
fragments. In particular, our results show that merging $\mathit{1CNF}$
formulas via the Hamming-distance based operator $\Delta^{H,\sum}$
does not need additional care, since the result is guaranteed to stay
in the fragment.

\paragraph{Related Work.}
Previous work on merging in fragments of propositional logic proposed
an adaptation of existing belief merging operators to ensure that the
result of merging belongs to a given fragment \cite{CreignouPRW14}, or
modified the rationality postulates in order to function in the
$\mathit{Horn}$ fragment \cite{HaretRW15}.  Our approach is different, since
we do not require that the result of merging stays in a given
fragment.  On the contrary, we want to decompose arbitrary bases into
a fragment-profile.  Recent work by Liberatore has also addressed a
form of meta-reasoning over belief change operators.  In
\cite{Liberatore:2015}, the input is a profile of knowledge bases with
the expected result of merging $R$, and the aim is to determine the
reliability of the bases (for instance, represented by weights) which
allow the obtaining of $R$.
In another paper,
\citeauthor{Liberatore15}
\shortcite{Liberatore15}
identifies, given a sequence of belief revisions and their results, the
initial pre-order which characterizes the revision operator.  Finally,
even if our approach may seem related to {\em Knowledge Compilation}
(KC) \cite{DarwicheM02,FargierM14,Marquis15}, both methods are in fact
conceptually different. KC aims at modifying a knowledge base $K$ into
a knowledge base $K'$ such that the most important queries for a given
application (consistency checking, clausal entailment, model
counting, $\dots$) are simpler to solve with $K'$. Here, we are interested
in the extent to which it is possible to \emph{equivalently represent} an arbitrary knowledge base
by simpler fragments when using merging as a recovery operation.

\paragraph{Main Contributions.}
We formally introduce the concept of knowledge distributability, as well
as a restricted version of it where the profile is limited to a single
knowledge base (simplifiability). We show that for drastic distance
arbitrary knowledge can be distributed into bases restricted to mostly any kind of
fragment, while simplifiability is limited to trivial cases. On the
other hand, for Hamming-distance based merging the picture is more
opaque.  We show that for $\mathit{1CNF}$, distributability w.r.t.\
$\Delta^{H,\sum}$ is limited to trivial cases, while slightly more can
be done with $\Delta^{H,{\mathit{GMin}}}$ and $\Delta^{H,{\mathit{GMax}}}$.  For $\mathit{2CNF}$
we show that arbitrary knowledge can be distributed and even be
simplified.  Finally, we discuss the $\mathit{Horn}$ fragment for which the
results for $\Delta^{H,\sum}$, $\Delta^{H,{\mathit{GMin}}}$ and
$\Delta^{H,{\mathit{GMax}}}$ are situated in between the two former fragments.

\section{Background}

\paragraph{Fragments of Propositional Logic.}
We consider $\L$ as the language of propositional logic
over some fixed alphabet $\mathcal{U}$ of propositional atoms.
We use standard connectives $\vee$, $\wedge$, $\neg$,
and constants $\top$, $\bot$. A clause is a disjunction of literals.
A clause is called \emph{Horn} if at most one of its literals is positive.
An interpretation is a set of atoms (those set to true).
The set of all interpretations is $2^{\mathcal{U}}$.
Models of a formula $\phi$ are denoted by $\mod(\phi)$. A knowledge base (KB) is a finite set of formulas and we identify models of a KB $K$ via
$\mod(K)=\bigcap_{\phi\in K}\mod(\phi)$. A profile is a finite non-empty tuple of KBs.
Two formulae $\phi_1, \phi_2$ (resp. KBs $K_1,K_2$) are equivalent, denoted $\phi_1 \equiv \phi_2$ (resp. $K_1 \equiv K_2$), when they have the same set of models.

We use a rather general and abstract notion of fragments.

\begin{definition}
A mapping $\cl{}: 2^{2^\mathcal{U}}\longrightarrow 2^{2^\mathcal{U}}$ is called \emph{closure-operator}
if it satisfies the following
for any ${\mathcal M},{\mathcal N}\subseteq 2^\mathcal{U}$:
\begin{itemize}
  \item If ${\mathcal M}\subseteq {\mathcal N}$, then
$\cl{}({\mathcal M})\subseteq\cl{}({\mathcal N})$
\item If $\vert {\mathcal M}\vert =1$, then $\cl{}({\mathcal M})={\mathcal M}$
\item $\cl{}(\emptyset)=\emptyset$.
\end{itemize}
\end{definition}

\begin{definition}
$\L'\subseteq \L$ is called a \emph{fragment} if
it is closed under conjunction (i.e.,
$\phi \wedge \psi \in \L'$
for any $\phi, \psi \in \L'$),
and there exists an associated closure-operator $\cl{}$ such that
(1) for all $\psi \in \L'$, $\mod(\psi) = \cl{}(\mod(\psi))$ and
(2) for all ${\mathcal M}\subseteq 2^\mathcal{U}$
there is a $\psi \in \L'$  with $\mod(\psi)= \cl{}({\mathcal M})$.
We often denote the closure-operator $\cl{}$ associated to a fragment $\L'$ as
$\cl{\L'}$.
\end{definition}

\begin{definition}
For a fragment $\L'$, we call
a finite set $K\subseteq \L'$ an
$\L'$-knowledge base.
An $\L'$-profile is a profile over $\L'$-knowledge bases.
A KB $K'\subseteq \L$ is called $\L'$-\emph{expressible} if there exists an $\L'$-KB $K$,
such that $K'\equiv K$.
\end{definition}

Many well known fragments of propositional logic
are indeed captured by our notion.
For the Horn-fragment $\L_{\mathit{Horn}}$, i.e.\ the set of all conjunctions of Horn clauses over $\mathcal{U}$,
take
the operator
$\cl{\L_{\mathit{Horn}}}$ defined as the fixed point of the function
$$
\cl{\L_{\mathit{Horn}}}^1({\mathcal M}) =
\{ \omega_1\cap \omega_2 \mid \omega_1,\omega_2\in {\mathcal M} \}.
$$
The fragment $\L_{\mathit{Krom}}$ which is restricted to formulas over clauses
of length at most $2$ is linked to the operator
$\cl{\L_{\mathit{Krom}}}$ defined as the fixed point of the function $\cl{\L_{\mathit{Krom}}}^1$ given by
$$
\cl{\L_{\mathit{Krom}}}^1({\mathcal M}) =
\{  \mathrm{maj}_3(\omega_1,\omega_2,\omega_3) \mid \omega_1,\omega_2,\omega_3\in {\mathcal M} \}.
$$
Here, we use the ternary majority function $\mathrm{maj}_3(\omega_1,\omega_2,\omega_3)$ which yields an interpretation
containing those atoms which are true in at least two out of $\omega_1,\omega_2,\omega_3$.
Finally, we are also interested in the $\L_{\mathit{1CNF}}$ fragment which is just composed of conjunctions of literals;
its associated operator $\cl{\L_{\mathit{1CNF}}}$ is defined as the fixed point of the function
\begin{eqnarray*}
\cl{\L_{\mathit{1CNF}}}^1({\mathcal M}) & = &
\{ \omega_1\cap \omega_2,\omega_1\cup  \omega_2 \mid \omega_1,\omega_2\in {\mathcal M} \} \cup\\
&& \{ \omega_3 \mid \omega_1\subseteq \omega_3 \subseteq \omega_2; \omega_1,\omega_2\in{\mathcal M}\}.
\end{eqnarray*}
Note that full classical logic is given via the identity closure operator $\cl{\L}({\mathcal M})={\mathcal M}$.

\paragraph{Merging Operators.}
We focus on {\em IC-merging},
where a profile
is mapped into a KB, such that
the result
satisfies some integrity
constraint.
Postulates for IC-merging have been stated
in \cite{KP02}.
We recall a specific family of  IC-merging operators,
based on distances between  interpretations, see also
\cite{KLM04}.

\begin{definition}
A distance between interpretations is a mapping $d$ from two
interpretations to a non-negative real number,
such that for
all $\omega_1,\omega_2,\omega_3\subseteq \mathcal{U}$,
(1) $d(\omega_1,\omega_2) = 0$ iff $\omega_1 = \omega_2$;
(2) $d(\omega_1,\omega_2) = d(\omega_2,\omega_1)$; and
(3) $d(\omega_1,\omega_2) +
    d(\omega_2,\omega_3) \geq d(\omega_1,\omega_3)$.
We will use two specific distances:
\begin{description}
  \item[drastic distance] $D(\omega_1,\omega_2) = 1$ if $\omega_1 =
    \omega_2$, $0$ otherwise;
  \item[Hamming distance] $H(\omega_1,\omega_2) = |(\omega_1 \setminus
    \omega_2) \cup (\omega_2 \setminus \omega_1)|$.
\end{description}
\end{definition}

We overload the previous notations to define the distance between
an interpretation $\omega$ and a KB $K$: if $d$ is a distance between interpretations,
then
\[
d(\omega,K) = \min_{\omega' \in \mod(K)}d(\omega,\omega').
\]
Next, an aggregation function must be used to
evaluate the distance between an interpretation and a profile.

\begin{definition}\label{def:aggregation-function}
An aggregation function
$\otimes$
associates a
non-negative number to every finite tuple of non-negative numbers,
such that:
\begin{enumerate}
  \item
If $x \leq y$, then
    $\otimes(x_1,\dots,x,\dots,x_n) \leq \otimes(x_1,\dots,y,\dots,x_n)$;
  \item
$\otimes(x_1,\dots,x_n) = 0$ iff $x_1 =\dots = x_n
    = 0$;
  \item
For every non-negative number $x$, $\otimes(x) = x$.
\end{enumerate}
As aggregation functions, we will consider the sum $\sum$, and $\GMAX$
and $\GMIN$\footnote{$\GMAX$ and $\GMIN$ are also known as ${\mathit{leximax}}$ and ${\mathit{leximin}}$ respectively. Stricto sensu, these functions return a vector of numbers, and not a single number. However, $\GMAX$ (resp. $\GMIN$) can be associated with an aggregation function as defined in Definition~\ref{def:aggregation-function} which yields the same vector ordering than $\GMAX$ (resp. $\GMIN$). We do a slight abuse by using directly $\GMAX$ and $\GMIN$ as the names of aggregation functions. See \cite{KLM02}.}, defined as follows. Given a profile $(K_1,\dots,K_n)$,
let $V_\omega = (d_1^\omega,\dots,d_n^\omega)$ be the vector of
distances s.t. $d_i^\omega =
d(\omega,K_i)$. $\GMAX(d_1^\omega,\dots,d_n^\omega)$
(resp. $\GMIN(d_1^\omega,\dots,d_n^\omega)$) is defined by ordering
$V_\omega$ in decreasing (resp. increasing) order. Given two
interpretations $\omega_1,\omega_2$, $\GMAX(d_1^{\omega_1},\dots,d_n^{\omega_1})
\leq \GMAX(d_1^{\omega_2},\dots,d_n^{\omega_2})$
(resp. $\GMIN(d_1^{\omega_1},\dots,d_n^{\omega_1}) \leq
\GMIN(d_1^{\omega_2},\dots,d_n^{\omega_2})$) is defined by comparing them
w.r.t. the lexicographic ordering.
\end{definition}

Finally,
let $d$ be a distance, $\omega$ an interpretation
and $E = (K_1,\dots,K_n)$ a profile. Then,
\[
d^\otimes(\omega,E) = \otimes(d(\omega,K_1),\dots,d(\omega,K_n)).
\]
If there is no ambiguity about the aggregation function $\otimes$, we write $d(\omega,E)$ instead of $d^\otimes(\omega,E)$.

\begin{definition}\label{def:icm}
For any distance $d$ between interpretations, and any aggregation
function $\otimes$, the merging operator $\Delta^{d,\otimes}$ is a
  mapping from a profile $E$ and a formula $\mu$ to a KB, such that
\[
\mod(\Delta^{d,\otimes}_\mu(E))=\min(\mod(\mu),\leq_E^{d,\otimes}),
\]
with $\omega_1 \leq_E^{d,\otimes} \omega_2$ iff $d^\otimes(\omega_1,E) \leq d^\otimes(\omega_2,E)$.
\end{definition}

When we consider a profile containing a single knowledge base $K$, all aggregation functions are equivalent; we write $\Delta_\mu^{d}(K)$ instead of $\Delta_\mu^{d,\otimes}((K))$ for readability.
For drastic distance, ${\mathit{GMin}}$, ${\mathit{GMax}}$, and $\sum$ are equivalent for arbitrary profiles. Thus,
whenever we show results
for $\Delta^{D,\sum}$, these carry over to
$\Delta^{D,{\mathit{GMin}}}$ and
$\Delta^{D,{\mathit{GMax}}}$.

\section{Main Concepts and General Results}

We now give the central definition for a knowledge base being distributable
into a profile from a certain fragment with respect to a given merging operator.

\begin{definition}
Let $\Delta$ be a merging operator, $K\subseteq \L$ be an arbitrary KB, and $\L'$ be a fragment.
$K$ is called $\L'$-\emph{distributable} w.r.t.\ $\Delta$
if there exists an $\L'$-profile $E$ and a formula $\mu\in\L'$,
such that $\Delta_\mu(E) \equiv K$.
\end{definition}

\begin{example}\label{ex:1}
Let $\mathcal{U}=\{a,b\}$ and consider
$K=\{ a\vee b \}$ which we want to
check for $\L_{\mathit{Horn}}$-distributability w.r.t.\ operator $\Delta^{H,\sum}$.
We have $\mod(K)=\{ \{a\},\{b\},\{a,b\}\}$, thus $K$ is not
$\L_{\mathit{Horn}}$-expressible
(note that
$\cl{\L_{\mathit{Horn}}}(\mod(K))=\{\emptyset,\{a\},\{b\},\{a,b\}\}\neq\mod(K)$),
otherwise $K$ would be distributable in a simple way (see Proposition~\ref{prop:clear} below).

Take the $\L_{\mathit{Horn}}$-profile
$E=(K_1,K_2)$ with
$K_1 = \{ a \wedge b \}$,
$K_2 = \{ \neg a \vee \neg b \}$,
together with the empty constraint $\mu=a \vee \neg a$.
We have
$\mod(K_1)=\{\{a,b\}\}$,
$\mod(K_2)=\{\{a\},\{b\},\emptyset\}$.
In the following matrix, each line corresponds to the distance between a model of $\mu$ and a KB from the profile $E$ (columns $K_1$ and $K_2$), or between a model of $\mu$ and the profile
using the sum-aggregation over the distances to the single KBs
(column $\sum$).
\[
\begin{array}{clllll}
& K_1 & K_2  & \sum\\
\{a,b\}	 	        &	0	&		1   &  1\\
\{a\}           	&	1	&		0   &  1\\
\{b\}           	&	1	&		0   &  1\\
\emptyset             	&	2	&		0   &  2\\
\end{array}
\]
We observe that $\mod(\Delta^{H,\sum}_\mu(E))=\{ \{a\},\{b\},\{a,b\}\}$, thus
$\Delta^{H,\sum}_\mu(E) \equiv K$ as desired.
It is easily checked that also other aggregations work:
$\Delta^{H,{\mathit{GMax}}}_\mu(E)  \equiv
\Delta^{H,{\mathit{GMin}}}_\mu(E)
\equiv K$.
\hfill$\diamond$
\end{example}

Next, we recall that IC-merging of a single KB yields revision.
Thus, the concept we introduce next is also of interest, as it represents
a certain form of reverse revision.

\begin{definition}
Let $\Delta$ be a merging operator, $K\subseteq \L$ an arbitrary KB,
and $\L'$ a fragment.
$K$ is called $\L'$-\emph{simplifiable} w.r.t.\ $\Delta$
if there exists an $\L'$-KB $K'$  and $\mu\in\L'$,
such that $\Delta_\mu(K') \equiv K$.
\end{definition}

As we will see later,
the KB $K$ from Example~\ref{ex:1} cannot be $\L_{\mathit{Horn}}$-simplified w.r.t.\ $\Delta^{H}$; in other words,
we need here at least two KBs to ``express'' $K$.
However, it is rather straightforward that any $\L'$-expressible KB can be $\L'$-simplified.

\begin{proposition}\label{prop:clear}
For every fragment $\L'$ and
every KB $K$,
it holds that $K$ is $\L'$-simplifiable (and thus also $\L'$-distributable) w.r.t.\ $\Delta$,
whenever $K$ is $\L'$-expressible.
\end{proposition}
\begin{proof}
Let $K'$ be an $\L'$-KB equivalent to $K$,
and let $\mu=(\bigwedge_{\phi \in K'} \phi)$.
Thus, $\mu \in\L'$ by definition of fragments and it is easily verified
that $\Delta_\mu(K') \equiv K$.
\end{proof}

Next, we show that
in order
to determine whether a KB $K$ is $\L'$-distributable,
it is sufficient to consider constraints $\mu$ such that
$\mod(\mu)=\cl{\L'}(\mod(K))$.

\begin{proposition}\label{prop:mu}
  Let $K \in \L$ be a KB, $\L'$ be a
  fragment, $E$ an $\L'$-profile and $\mu\in\L'$.  Then
	$\Delta_\mu(E)\equiv K$
	implies
	$\Delta_{\mu'}(E)\equiv K$ for any $\mu'$ such that
 	$\mod(\mu') = \cl{\L'}(\mod(K))$.
\end{proposition}

\begin{proof}
Let $\Delta=\Delta^{d,\otimes}$.
By Definition~\ref{def:icm},
$\mod(K) = \min(\mod(\mu),\leq^{d,\otimes}_E)$,
hence
$\mod(K) \subseteq \mod(\mu)$.
Moreover, $\mu$ is $\L'$-closed, so
$\cl{\L'}(\mod(K))=\mod(\mu')
\subseteq \mod(\mu)$.
We get $\mod(K)\subseteq \mod(\mu') \subseteq \mod(\mu)$.
Thus,
$\mod(K) = \min(\mod(\mu'),\leq^{d,\otimes}_E)$,
i.e.\
	$\Delta_{\mu'}(E)\equiv K$.
\end{proof}

Next, we give two positive results for distributing knowledge
in any fragment. The key idea is to use KBs in the profile
which have exactly one model (our notion of fragment guarantees existence
of such KBs). The first result is independent of the distance notion but requires
${\mathit{GMin}}$ as the aggregation function. The second result is for drastic distance and  thus
works for any of the aggregation functions we consider.

\begin{theorem}\label{thm:gmin}
Let $d$ be a distance and
$\L'$  be a fragment.
Then for
every KB $K$, such that for
  all distinct $\omega_1,\omega_2 \in \mod(K)$, $d(\omega_1,\omega_2)=e$ for some
  $e>0$, it holds that $K$ is $\L'$-distributable w.r.t. $\Delta^{d,\GMIN}.$
\end{theorem}

\begin{proof}
  Build the $\L'$-profile $E$ such that for each $\omega \in \mod(K)$,
  there is a KB with $\omega$ as its only model.  Thus
  all models of $K$ get a ${\mathit{GMin}}$-vector $(0,e,e,e,e,\ldots)$.  All
  interpretations from $\cl{\L'}(\mod(K))\setminus \mod(K)$ get a
  vector $(f,g,\ldots)$ with $f>0$. Hence, we have
  $\min(\mod(\mu),\leq_E^{d,\GMIN}) = \mod(K)$ using $\mu\in\L'$ with
$\mod(\mu) = \cl{\L'}(\mod(K))$.
\end{proof}

\begin{theorem}
For every fragment $\L'$ and
every knowledge base $K$,
it holds that $K$ is $\L'$-distributable w.r.t.\ $\Delta^{D,\oplus}$, for $\oplus \in \{\sum,\GMIN,\GMAX\}$.
\end{theorem}
\begin{proof}
  Given a fragment $\L'$, we take $E=\{ K_\omega \mid \omega\in
  \mod(K) \}$ where $K_\omega\in \L'$ is a knowledge base with single
  model $\omega$ (such $K_\omega\in \L'$ exists due to our definition
  of fragments), and let $\mu$ be such that
  $\mod(\mu)=\cl{\L'}(\mod(K))$; hence also $\mu\in\L'$.  Let
  $\omega'\in\mod(\mu)$ and $n=|\mod(K)|$, we observe that
  $\sum_{K_\omega \in E} H(\omega',K_\omega) = n-1$ when $\omega'\in
  \mod(K)$, and $n$ otherwise.  Thus, $\Delta_\mu^{D,\sum}(E) \equiv
  K$. The same result holds for $\Delta_\mu^{D,\GMAX}$ and $\Delta_\mu^{D,\GMIN}$.
\end{proof}

Concerning simplifiability w.r.t.\ drastic distance based operators,
Proposition
\ref{prop:clear} cannot be improved.

\begin{theorem}\label{thm:d}
For every fragment $\L'$ and
every KB $K$,
$K$ is $\L'$-\emph{simplifiable} w.r.t.\ $\Delta^D$
iff $K$ is $\L'$-expressible.
\end{theorem}
\begin{proof}
The if-direction is by Proposition \ref{prop:clear}.
For the other direction,
suppose $K$ is not $\L'$-expressible.
We show that for any $\L'$-KB $K'$,
$\Delta^D_\mu(K')\not\equiv K$ with
$\mu=\cl{\L'}(K)$. By Proposition~\ref{prop:mu} the result then follows.
Now suppose there exists an $\L'$-KB $K'$ such that
$\Delta^D_\mu(K') \equiv K$.
First observe that since $K$ is not $\L'$-expressible,
$\mod(\mu)\supset \mod(K)$.
Since
we are working with drastic distance, in order to promote models of $K$,
we also need them in $K'$, hence $\mod(K')\supseteq \mod(K)$ and since $K'$ is from $\L'$
we have $\mod(K')\supseteq \cl{\L'}(K) = \mod(\mu)$.
Thus there exists $\omega\in\cl{\L'}(\mod(K))\setminus \mod(K)$ having distance $0$ to $K'$,
and thus $\omega\in\Delta^D_\mu(K')$. Since $\omega\notin\mod(K)$, this yields
a contradiction to
$\Delta^D_\mu(K') \equiv K$.
\end{proof}

\section{Hamming Distance and Specific Fragments}
We first consider the simplest fragment under consideration, namely
conjunction of literals. As it turns out, (non-trivial) distributability
for this
fragment w.r.t.\ $\Delta^{H,\Sigma}$ is not achievable.
We then see that more general fragments allow for
non-trivial distributions. In particular, we show that every
KB is distributable (and even simplifiable) in the
$\mathit{2CNF}$ case,
and we finally give a few
observations
for $\L_{\mathit{Horn}}$.

\subsection{The 1CNF Fragment}

The following technical result is important to prove the main result in this section.

\begin{lemma}\label{cor:cap-cup-equality}
	For any $\L_{\mathit{1CNF}}$-profile $E = (K_1, \dots, K_n)$
and interpretations $\omega_1, \omega_2$, it holds that:
		$$
			\distance{\omega_1}{E} + \distance{\omega_2}{E} = \distance{\omega_1 \cap \omega_2}{E} + \distance{\omega_1 \cup \omega_2}{E}.
		$$
\end{lemma}
\begin{proof}

	It suffices to show that for each $K_i$ in profile $E$,
	$
	\distance{\omega_1}{K_i} + \distance{\omega_2}{K_i} = \distance{\omega_1\cap \omega_2}{K_i} + \distance{\omega_1 \cup \omega_2}{K_i}
	$.

	Indeed,
	summing up these equalities over all $K_i \in E$, we get
	\begin{eqnarray*}
	&\sum_{K_i \in E}\distance{\omega_1}{K_i} + \sum_{K_i \in E}\distance{\omega_2}{K_i} =\\
	&\sum_{K_i \in E}\distance{\omega_1\cap \omega_2}{K_i} + \sum_{K_i \in E}\distance{\omega_1 \cup \omega_2}{K_i}.
	\end{eqnarray*}
	Since $\distance{\omega}{E} = \Sigma_{K_i \in E} \distance{\omega}{K_i}$, for any interpretation $\omega$, our conclusion then follows immediately.

Thus,
take $\omega_1', \omega_2'$ to be two interpretations that are closest to $\omega_1$ and $\omega_2$, respectively, among the models of $\mod(K_i)$.
In other words, $H(\omega_1, \omega_1') = \min_{\omega \in \mod(K_i)} H(\omega_1, \omega)$ and $H(\omega_2, \omega_2') = \min_{\omega \in \mod(K_i)} H(\omega_2, \omega)$.
By induction on the number of propositional atoms in $\L$, we can show that $\omega_1' \cap \omega_2'$ and $\omega_1' \cup \omega_2'$ are closest in $\mod(K_i)$ to $\omega_1 \cap \omega_2$ and $\omega_1 \cup \omega_2$, respectively.
Thus, we have that $\distance{\omega_1}{K_i} = H(\omega_1, \omega_1')$, $\distance{\omega_2}{K_i} = H(\omega_2, \omega_2')$, $\distance{\omega_1 \cap \omega_2}{K_i} = H(\omega_1 \cap \omega_2, \omega_1' \cap \omega_2')$, $\distance{\omega_1 \cup \omega_2}{K_i} = H(\omega_1 \cup \omega_2, \omega_1' \cup \omega_2')$,
and our problem reduces to showing that 	$H(\omega_1, \omega_1') + H(\omega_2, \omega_2') = H(\omega_1 \cap \omega_2, \omega_1' \cap \omega_2') + H(\omega_1 \cup \omega_2, \omega_1' \cup \omega_2')$.
By using induction on the number of propositional atoms in $\L$ again, we can show that this equality holds. The argument runs as follows: in the base case, when the alphabet consists of just one propositional atom, the equality is shown to be true by checking all the cases.
For the inductive step we assume the claim holds for an alphabet of size $n$ and show that it also holds for an alphabet of size $n+1$.
More concretely, we analyze the way in which the Hamming distances between interpretations change when we add a propositional atom to the alphabet.
An analysis of all the possible cases shows that the equality holds.
\end{proof}

Next we observe certain patterns of interpretations that indicate whether a KB is $\L_{\mathit{1CNF}}$-expressible or not.

\begin{definition}\label{def:critical-pair}
If $K$ is a knowledge base, then a pair of interpretations $\omega_1$ and $\omega_2$ are called \textit{critical with respect to $K$} if $\omega_1 \nsubseteq \omega_2$ and $\omega_2 \nsubseteq \omega_1$, and one of the following cases holds:
	\begin{enumerate}
	\item $\omega_1, \omega_2 \in \mod(K)$ and $\omega_1 \cap \omega_2, \omega_1 \cup \omega_2 \notin \mod(K)$,
	\item $\omega_1, \omega_2, \omega_1 \cap \omega_2 \in \mod(K)$ and $\omega_1 \cup \omega_2 \notin \mod(K)$,
	\item $\omega_1, \omega_2, \omega_1 \cup \omega_2 \in \mod(K)$ and $\omega_1\cap \omega_2 \notin \mod(K)$,
	\item $\omega_1 \cap \omega_2, \omega_1 \cup \omega_2 \in \mod(K)$ and $\omega_1, \omega_2 \notin \mod(K)$, or
	\item $\omega_1, \omega_1 \cap \omega_2, \omega_1 \cup \omega_2 \in \mod(K)$ and $\omega_2 \notin \mod(K)$.
	\end{enumerate}
\end{definition}

\begin{lemma}\label{lemma:critical-pairs}
	If a KB $K$ is not $\L_{\mathit{1CNF}}$-expressible, then there exist
$\omega_1, \omega_2 \in \cl{\L_{\mathit{1CNF}}}(K)$ being critical with respect to $K$.
\end{lemma}
\begin{proof}
	The fact that $K$ is not $\L_{\mathit{1CNF}}$-expressible implies that either: (i) $K$ is not closed under intersection or union, or (ii) there are $w_1, w_2, w_3 \in \cl{\L_{\mathit{1CNF}}}(K)$ such that $w_1 \subseteq w_3 \subseteq w_2$, and $w_1, w_2 \in \mod(K)$, $w_3 \notin \mod(K)$. Case (i) implies that there exist $w_1, w_2 \in \mod(K)$ such that one of Cases 1-3 from Definition~\ref{def:critical-pair} holds. If we are in Case (ii), then consider the interpretation $w_4 = (w_2 \backslash w_3) \cup w_1$. Clearly, $w_1 \subseteq w_4 \subseteq w_2$, hence $w_4 \in \cl{\L_{\mathit{1CNF}}}(K)$. Also, $w_3 \cap w_4 = w_1$ and $w_3 \cup w_4 = w_2$. There are two sub-cases to consider here. If $w_4 \notin \mod(K)$, then we are in Case~4 of Definition~\ref{def:critical-pair}. If $w_4 \in \mod(K)$, then we are in Case~5 of Definition~\ref{def:critical-pair}.
\end{proof}

\begin{example}
Let us consider the KB $K$ such that $\mod(K) = \{\emptyset,\{a\},\{b\},\{c\},\{a,c\},\{b,c\},\{a,b,c\}\}$. $K$ is not $\mathit{1CNF}$-expressible; indeed, $\cl{\mathit{1CNF}}(\mod(K)) = \mod(K) \cup \{\{a,b\}\}$.

Here, we identify several sets of critical interpretations w.r.t. $K$.
First, $S_1 = \{\{a,c\}, \{a,b\}, \{a\}, \{a,b,c\}\}$ corresponds to the situation described in Case 5 of Definition 9, with $\omega_1 = \{a,c\}$ and $\omega_2 = \{a,b\}$.

The set $S_2 = \{\{b,c\},\{a,b\}, \{b\}, \{a,b,c\}\}$ also corresponds to Case 5, with $\omega_1 = \{b,c\}$ and $\omega_2 = \{a,b\}$.

We can also consider the set of interpretations $S_3 = \{\emptyset, \{a\}, \{b\}, \{a,b\}\}$, which corresponds to Case 2 of Definition 9, with $\omega_1 = \{a\}$ and $\omega_2 = \{b\}$.
The models of $K$ and the sets of critical interpretations are represented in Figure~\ref{fig:critical-interpretations}.

\begin{figure}[h]
\centering
\subfloat[$S_1$]{
\scalebox{0.8}{
\begin{tikzpicture}[auto,node distance=1cm,thick]
\node (abc) {$a,b,c$};
\node (ab) [below of=abc] {$a,b$};
\node (ac) [left of=ab] {$a,c$};
\node (bc) [right of=ab] {$b,c$};
\node (a) [below of=ac] {$a$};
\node (b) [below of=ab] {$b$};
\node (c) [below of=bc] {$c$};
\node (empty) [below of=b] {$\emptyset$};

 \fill[color=lightgray,opacity=0.5,rounded corners=5pt,even odd rule] (a.west) -- (ac.north west) -- (abc.north west) -- (abc.north)
              -- (abc.north east) -- (bc.north east) -- (c.south east)
              -- (empty.south east) -- (empty.south)
              -- (empty.south west) -- (a.south west) -- cycle
              (abc.south) -| (bc.west) |- (b.north) -| (ac.east) |- (abc.south);

\draw[loosely dashed,rounded corners=5pt] (a.south east) -- (ab.south east) -- (ab.north east)
                      -- (abc.north east) -- (abc.north west)
                      -- (ac.north west) -- (ac.south west) -- (a.south west)
                      -- cycle;

\end{tikzpicture}
}}
\subfloat[$S_2$]{
\scalebox{0.8}{
\begin{tikzpicture}[auto,node distance=1cm,thick]
\node (abc) {$a,b,c$};
\node (ab) [below of=abc] {$a,b$};
\node (ac) [left of=ab] {$a,c$};
\node (bc) [right of=ab] {$b,c$};
\node (a) [below of=ac] {$a$};
\node (b) [below of=ab] {$b$};
\node (c) [below of=bc] {$c$};
\node (empty) [below of=b] {$\emptyset$};

 \fill[color=lightgray,opacity=0.5,rounded corners=5pt,even odd rule] (a.west) -- (ac.north west) -- (abc.north west) -- (abc.north)
              -- (abc.north east) -- (bc.north east) -- (c.south east)
              -- (empty.south east) -- (empty.south)
              -- (empty.south west) -- (a.south west) -- cycle
              (abc.south) -| (bc.west) |- (b.north) -| (ac.east) |- (abc.south);

\draw[loosely dashed,rounded corners=5pt]  (b.west)
              -- (abc.west) |- (abc.north east)
              -- (bc.north east) -- (bc.south east)
              -- (b.south east) -- (b.south west) -- cycle;
\end{tikzpicture}
}}
\subfloat[$S_3$]{
\scalebox{0.8}{
\begin{tikzpicture}[auto,node distance=1cm,thick]
\node (abc) {$a,b,c$};
\node (ab) [below of=abc] {$a,b$};
\node (ac) [left of=ab] {$a,c$};
\node (bc) [right of=ab] {$b,c$};
\node (a) [below of=ac] {$a$};
\node (b) [below of=ab] {$b$};
\node (c) [below of=bc] {$c$};
\node (empty) [below of=b] {$\emptyset$};

 \fill[color=lightgray,opacity=0.5,rounded corners=5pt,even odd rule] (a.west) -- (ac.north west) -- (abc.north west) -- (abc.north)
              -- (abc.north east) -- (bc.north east) -- (c.south east)
              -- (empty.south east) -- (empty.south)
              -- (empty.south west) -- (a.south west) -- cycle
              (abc.south) -| (bc.west) |- (b.north) -| (ac.east) |- (abc.south);

\draw[loosely dashed,rounded corners=5pt]  (a.west) -- (ab.north west)
              -- (ab.north east)
              -- (empty.south east) -- (empty.south)
              -- (empty.south west) -- (a.south west) -- cycle;
\end{tikzpicture}
}}
\caption{Models of $K$ are in the shaded area; critical interpretations are in the dashed areas. \label{fig:critical-interpretations}}
\end{figure}
\end{example}

We can now state the central result of this section.

\begin{theorem}
A KB $K$ is $\L_{\mathit{1CNF}}$-distributable with respect to $\Delta^{H, \Sigma}$ if and only if $K$ is $\L_{\mathit{1CNF}}$-expressible.
\end{theorem}
\begin{proof}
\textit{If part.} By Proposition~\ref{prop:clear}.

\textit{Only if part.} Let $K$ be a KB that is not $\L_{\mathit{1CNF}}$-expressible. We will show that it is not $\L_{\mathit{1CNF}}$-distributable w.r.t.\ $\Delta^{H,\Sigma}$. Suppose, on the contrary, that $K$ is $\L_{\mathit{1CNF}}$-distributable. Then there exists an $\L_{\mathit{1CNF}}$ profile $E = (K_1, \dots, K_n)$ such that $\merge{H, \Sigma}{\mu}{E} \equiv K$, where $\mod(\mu) = \cl{\L_{\mathit{1CNF}}}(\mod(K))$ (cf.\ Proposition~\ref{prop:mu}).

By Lemma~\ref{lemma:critical-pairs}, there exist interpretations $\omega_1, \omega_2 \in \mod(\mu)$ that are critical with respect to $K$. By Lemma~\ref{cor:cap-cup-equality}, we have
\begin{equation}\label{eq:cap-cup-eq}
\!
	\distance{\omega_1}{E} {+} \distance{\omega_2}{E} {=} \distance{\omega_1 {\cap} \omega_2}{E} {+} \distance{\omega_1 {\cup} \omega_2}{E}.\!
\end{equation}
Let us now do a case analysis depending on the type of critical pair we are dealing with. If we are in Case~1 of Definition~\ref{def:critical-pair}, then it needs to be the case that $\distance{\omega_1}{E} = \distance{\omega_2}{E} = m$, $\distance{\omega_1 \cap \omega_2}{E} = m+k_1$ and $\distance{\omega_1 \cup \omega_2}{E}=m+k_2$, for some integers $m \geq 0$ and $k_1, k_2 > 0$. Plugging these numbers into Equality~(\ref{eq:cap-cup-eq}), we get that $2m = 2m + k_1 + k_2$ and $k_1 + k_2 = 0$. Since $k_1, k_2 > 0$, we have arrived at a contradiction.
If we are in Case~2, then it needs to be the case that ${\distance{\omega_1 \cap \omega_2}{E}} = \distance{\omega_1 \cup \omega_2}{E} = m$, $\distance{\omega_1}{E} = m+k_1$ and $\distance{\omega_2}{E}=m+k_2$, for some integers $m \geq 0$ and $k_1, k_2 > 0$. Plugging these numbers into Equality~(\ref{eq:cap-cup-eq}) again, we get a contradiction along the same lines as in Case~1.
If we are in Case~3, then it needs to hold that $\distance{\omega_1}{E} = \distance{\omega_1 \cap \omega_2}{E} = \distance{\omega_1 \cup \omega_2}{E} = m$, $\distance{\omega_2}{E} = m+k$, for some integers $m \geq 0$ and $k > 0$. Plugging these numbers into Equality~(\ref{eq:cap-cup-eq}) gives us $2m + k = 2m$ and hence $k = 0$. Since $k > 0$, we have arrived at a contradiction. Cases~4 and~5 are entirely similar.
\end{proof}

In other words, for any $\L_{\mathit{1CNF}}$-profile and $\mu\in\mathit{1CNF}$,
$\Delta^{H,\sum}_\mu$ is guaranteed to be $\L_{\mathit{1CNF}}$-expressible as well.
As we have already shown in Theorem~\ref{thm:gmin}, this is not necessarily the case
if we replace $\sum$ by ${\mathit{GMin}}$.
The following example shows how to obtain a similar behavior for ${\mathit{GMax}}$; we then generalize
this idea below.

\begin{example}
Let
$\mathcal{U}=\{a,b\}$ and
$K=\{ a\vee b, \neg a \vee \neg b\}$.  We have
$\mod(K)=\{\{a\},\{b\}\}$. $K$ is not $\L_{\mathit{1CNF}}$-expressible, since
$\cl{\L_{\mathit{1CNF}}}(\mod(K))=2^\mathcal{U}$. Let $K_{S}$ be the $\L_{\mathit{1CNF}}$-KB with a single
model $S$ for any $S\subseteq \mathcal{U}$ and let us have a look at the following distance matrix
for $\mu$ with $\mod(\mu)=\cl{\L_{\mathit{1CNF}}}(\mod(K))$,
$E=(K_{\{a\}},K_{\{b\}})$,
and
$E'=(K_{\emptyset},K_{\{a,b\}})$.
\begin{small}
$$
\begin{array}{lcccccc}
&
\!\!K_{\emptyset} &
\!K_{\{a\}} &
\!K_{\{b\}} &
\!K_{\{a,b\}} &
\!\!H^{\mathit{GMin}}(E) &
\!\!H^{\mathit{GMax}}(E') \\
\emptyset & \!\!0 & 1 & 1 & 2 & \!(1,1) &\! (2,0) \\
\{a\} & \!\!1 &  0 & 2 & 1 & \!(0,2) & \!(1,1) \\
\{b\} & \!\!1 & 2 & 0 & 1 & \!(0,2) & \!(1,1) \\
\{a,b\} & \!\!2 & 1 & 1 & 0 & \!(1,1) & \!(2,0)
\end{array}
$$
\end{small}

Recall that the lexicographic order of the involved vectors is $(0,2)<(1,1)<(2,0)$.
We thus get that
$\Delta^{H,{\mathit{GMin}}}_\mu(E)\equiv K$ (see also Theorem~\ref{thm:gmin}), and
on the other hand,
$\Delta^{H,{\mathit{GMax}}}_\mu(E')\equiv K$.
\hfill$\diamond$
\end{example}

\begin{theorem}
Any KB $K$ such that $\mod(K)=\{\omega, \omega'\}$ is $\L_{\mathit{1CNF}}$-distributable with respect to $\Delta^{H, {\mathit{GMax}}}$.
\end{theorem}
\begin{proof}
If $K$ is $\L_{\mathit{1CNF}}$-expressible, then the conclusion follows from Proposition~\ref{prop:clear}.
If $K$ is not $\L_{\mathit{1CNF}}$-expressible, then consider the set $\cl{\L_{\mathit{1CNF}}}(\mod(K))\backslash\mod(K)=\{\omega_1, \dots, \omega_n\}$.
We define the profile $E=(K_1, \dots, K_n)$, where $\mod(K_i)=\{\mathcal{U}\backslash \omega_i\}$, for $i \in \{1, \dots, n\}$.
We show that $\merge{H, {\mathit{GMax}}}{\mu}{E} \equiv K$, where $\mod(\mu) = \cl{\L_{\mathit{1CNF}}}(\mod(K))$.

First,  we have that $H(\omega_i, \mathcal{U}\backslash \omega_i)=|\mathcal{U}|$, which implies that $H^{\mathit{GMax}}(\omega_i, E)={\mathit{GMax}}(|\mathcal{U}|, \dots)$, for any $i \in \{1, \dots, n\}$.
Furthermore, since $H(\omega, \mathcal{U} \backslash \omega_i) < |\mathcal{U}|$ and $H(\omega', \mathcal{U} \backslash \omega_i)<|\mathcal{U}|$, for any $i \in \{1, \dots, n\}$, it follows that $\omega <_E^{H, {\mathit{GMax}}} \omega_i$ and $\omega' <_E^{H, {\mathit{GMax}}} \omega_i$.
Next, we show that $H^{\mathit{GMax}}(\omega, E)=H^{\mathit{GMax}}(\omega', E)$.

Consider the vectors $V = (H(\omega, \omega_1), \dots, H(\omega, \omega_n))$ and $V' = (H(\omega', \omega_1), \dots, H(\omega', \omega_n))$.
Our claim is that ${\mathit{GMax}}(V)={\mathit{GMax}}(V')$.
To see why, notice that the elements in $\cl{\L_{\mathit{1CNF}}}(\mod(K))$ form a complete subset lattice with $\omega\cup \omega'$ and $\omega\cap \omega'$ as the top and bottom elements, respectively.
Let us write $H(\omega,\omega')=m$.
This lattice has $2^m$ elements, and the maximum distance of two elements in it is $m$.
Thus, the vector $V$ is the vector of distances between $\omega$ and every other element in this lattice, except itself and $\omega'$.
A similar consideration holds for $V'$.
Hence $V$ and $V'$ are vectors of length $2^{m-2}$ whose elements are $m-1,m-2, \ldots, 1$.
We can actually count how many times each number appears in $V$ and $V'$.
The number of interpretations in the lattice that are at distance of 1 from $\omega$ (and $\omega'$) is $\binom{1}{m}$: thus, $m-1$ appears $\binom{1}{m}$ times in $V$ (and $V'$).
The number of interpretations that are at distance 2 from $\omega$ (and $\omega'$) is $\binom{2}{m}$, thus $m-2$  appears  $\binom{2}{m}$ times in $V$ and $V'$.
We iterate this argument for every distance, up to 1.
It is then easy to see that, based on these considerations, $V$ and $V'$ are equal when sorted in descending order.
Our conclusion follows from this.
\end{proof}

\subsection{The 2CNF Fragment}

We show that every knowledge base $K$ can be distributed in the fragment $\L_{\mathit{Krom}}$. Even a single $\L_{\mathit{Krom}}$ knowledge base is enough to represent $K$.
Before giving the general result, we sketch the idea via an example.

\begin{example}
Let $K$ be a KB with
$\mod(K)=\{\{a,b\},\{b,c,e\},\{a,c,d\}\}$. We observe that $K$ is
not $\L_{\mathit{Krom}}$-expressible since $\cl{\L_{\mathit{Krom}}}(\mod(K))=\mod(K)\cup \{a,b,c\}$.
However, we can give an $\L_{\mathit{Krom}}$-KB $K'$ using three new atoms
$x,y,z$ to penalize the undesired interpretation $\{a,b,c\}$
such that $\Delta^H_\mu(K')\equiv K$,
with $\mu\in\L_{\mathit{Krom}}$ of the form $\mod(\mu)=\cl{\L_{\mathit{Krom}}}(\mod(K))$.
To this end, assume $K'$ with
$\mod(K') = \{ \omega_1,\omega_2,\omega_3,\omega_4\}$
of the form
\begin{eqnarray*}
\omega_1 &=& \{a,b,x,y\}, \\
\omega_2 &=& \{b,c,e,x,z\}, \\
\omega_3 &=& \{a,c,d,y,z\}, \\
\omega_4 &=& \{a,b,c,x,y,z\}.
\end{eqnarray*}
One can verify that $\cl{\L_{\mathit{Krom}}}(K')=\mod(K')$.
Thus, $K'$ can be picked from $\L_{\mathit{Krom}}$.
We use $\mu$ such that $\mod(\mu)=\cl{\L_{\mathit{Krom}}}(\mod(K))$ and get
distances
$$
\begin{array}{lcccccc}
&
\omega_1 &
\omega_2 &
\omega_3 &
\omega_4 &
\min
\\
\{a,b\} 	& 2 & 5 & 5 & 4 & 2
\\
\{b,c,e\} 	& 4 & 2 & 6 & 4 & 2
\\
\{a,c,d\} 	& 4 & 6 & 2 & 4 & 2
\\
\{a,b,c\}	& 3 & 4 & 4 & 3 & 3
\end{array}
$$
Here, each line gives the distance between a model of $\mu$ and a model of $K'$ ($\omega_i$ columns), or between a model of $\mu$ and $K'$ ($\min$ column).
The key observation is that pairs from $x,y,z$ as used in $\omega_1$, $\omega_2$, $\omega_3$
give minimal distances $2$ while the remaining interpretation $\omega_4$,
which corresponds to the closure of $K$, contains all three new
atoms (since $maj_3(\{x,y\},\{x,z\},\{y,z\})=\{x,y,z\}$).\hfill$\diamond$
\end{example}

\begin{theorem}\label{theorem:krom-representability}
Any KB
$K$
is $\L_{\mathit{Krom}}$-simplifiable w.r.t.~$\Delta_{\mu}^H$.
\end{theorem}

\begin{proof}
We have to show that
for any KB $K$, there exists an $\L_{\mathit{Krom}}$-KB $K'$ and a formula $\mu \in \L_{\mathit{Krom}}$ such that $\Delta_{\mu}^H(K') \equiv K$.
If $K$ is $\L_{\mathit{Krom}}$-expressible, the result is due to Proposition~\ref{prop:clear}. So suppose that $K$ is not $\L_{\mathit{Krom}}$-expressible and
let $\mod(K) = \{\omega_1,\dots,\omega_n\}$.
Consider a set of new atoms $A = \{a_1,\dots,a_n\}$, and for each
$\omega_i \in \mod(K)$, let $\omega_i' = \omega_i \cup A \setminus \{a_i\}$.
We define the $\L_{\mathit{Krom}}$-KB $K'$ and $\mu\in\L_{\mathit{Krom}}$ such that
\begin{eqnarray*}
\mod(K') & = & \cl{\L_{\mathit{Krom}}}(\{\omega_i' \mid \omega_i \in \mod(K)\}) \\
\mod(\mu) & = & \cl{\L_{\mathit{Krom}}}(\mod(K)).
\end{eqnarray*}
Let $\Omega'=\{\omega_i' \mid \omega_i \in \mod(K)\}$.
We first show that for each
$\omega\in\mod(K')\setminus\Omega'$,
$A\subseteq \omega$.
Indeed, for any triple
$\omega_j,\omega_k,\omega_l\in\mod(K)$, such that
$\omega_{jkl}=\mathrm{maj}_3(\omega_j,\omega_k,\omega_l)\notin\mod(K)$, we observe that
$\mathrm{maj}_3(\omega'_j,\omega'_k,\omega'_l)=
\omega_{jkl}\cup \mathrm{maj}_3(
A\setminus\{a_j\},
A\setminus\{a_k\},
A\setminus\{a_l\})=
\omega_{jkl}\cup A$.
Thus, for each
$\omega\in\cl{\L_{\mathit{Krom}}}^1(\Omega')\setminus\Omega'$,
$A\subseteq \omega$.
Recall that
$\mod(K')  =  \cl{\L_{\mathit{Krom}}}(\Omega')$.
It follows quite easily that
each further interpretation
$\omega\in\cl{\L_{\mathit{Krom}}}(\Omega')\setminus
(\cl{\L_{\mathit{Krom}}}^1(\Omega')\cup \Omega')$, also satisfies
$A\subseteq \omega$.

This shows that each model of $K'$ contains at least $n-1$ atoms from $A$.
Thus, for every model $\omega_i\in K$,
$H(\omega_i,K') = H(\omega_i,\omega_i') = n - 1$.
It remains to show that
for each
$\omega \in \mod(\mu) \setminus \mod(K)$,
$H(\omega,K') \geq n$.
First, let $\omega'\in\Omega'$.
Since $\omega \notin \mod(K)$, $\omega'\setminus A\neq\omega$
and since $\omega'$ contains $n-1$ elements from $A$,
we have
$H(\omega,\omega') \geq n$.
As shown above all other interpretations $\omega''\in
\mod(K')\setminus\Omega'$ contain all $n$ atoms from $A$,
thus
$H(\omega,\omega'') \geq n$, too.
\end{proof}

As an immediate consequence, we obtain that any KB $K$ is $\L_{\mathit{Krom}}$-distributable w.r.t.\ $\Delta^{H,\otimes}$ for \emph{any} aggregation function $\otimes$.
Note that this result is in strong contrast to the $\L_{\mathit{1CNF}}$ fragment,
where only $\L_{\mathit{1CNF}}$-expressible KBs are $\L_{\mathit{1CNF}}$-distributable w.r.t.\ $\Delta^{H,\sum}$.

\subsection{The Horn-Fragment}

We now turn our attention to the $\L_{\mathit{Horn}}$ fragment.
Recall Example~\ref{ex:1} where we
have shown how to distribute some non $\L_{\mathit{Horn}}$-expressible KB
using a profile over two $\L_{\mathit{Horn}}$-KBs.
Our first result shows that in this example case we cannot reduce to profiles of a single KB, i.e.\ that there are KBs which are
$\L_{\mathit{Horn}}$-distributable but not $\L_{\mathit{Horn}}$-simplifiable.

\begin{proposition}\label{prop:neg}
Given  a KB $K$ with $\mod(K)=\{\omega_1,\omega_2,\omega_3\}$, where $\omega_3=\omega_1\cup \omega_2$, $H(\omega_1, \omega_2)=2$ and $\omega_1,\omega_2$ are incomparable. Then $K$ is not $\L_{\mathit{Horn}}$-simplifiable w.r.t. $\Delta^H$.
\end{proposition}

\begin{proof}
The situation described in the Proposition corresponds to
$K = \{\omega \cup \{a\},\omega \cup \{b\}, \omega \cup \{a,b\}\}$ with $\omega$ some interpretation which does not contain $a$ or $b$.  We need $\mod(\mu)=\{\omega,\omega \cup \{a\},\omega \cup \{b\}, \omega \cup \{a,b\}\}$, as required by Proposition~\ref{prop:mu}.
We want to identify a $\L_{\mathit{Horn}}$-KB $K'$ such that $\Delta_\mu^H(K') \equiv K$. This means that $\omega$ is the single model of $\mu$ which is not minimal w.r.t. the Hamming distance. Let $\omega_1'$ be the model in $K'$ closest to $\omega_1 = \omega \cup \{a\}$ and $\omega_2'$ the one closest to $\omega_2 = \omega \cup \{b\}$. We need $a\in \omega_1'$ and $b\in \omega_2'$; otherwise $H(\omega,\omega_1')<H(\omega_1,\omega_1')$ or $H(\omega,\omega_2')<H(\omega_2,\omega_2')$;
further we need $b\notin \omega_1'$ and $a\notin \omega_2'$;
otherwise $H(\omega_3,\omega_1')<H(\omega_1,\omega_1')$ or $H(\omega_3,\omega_2')<H(\omega_2,\omega_2')$.
Hence $\omega_1'$ and $\omega_2'$ are incomparable thus also $\omega_1'\cap \omega_2' \in \mod(K')$, since $K'$ is a Horn KB. But then $H(\omega,\omega_1'\cap \omega_2') \leq H(\omega_1,\omega_1')$.
\end{proof}

Our next result shows that
$\Delta^H$ nonetheless increases the range of $\L_{\mathit{Horn}}$-simplifiable KBs compared to $\Delta^D$ (recall Theorem~\ref{thm:d}).

\begin{proposition}\label{thm:twomods}
Any knowledge base $K$ with $\mod(K)=\{\omega_1,\omega_2\}$
is $\L_{\mathit{Horn}}$-simplifiable w.r.t.\ $\Delta^{H}$.
\end{proposition}

\begin{proof}
  If $\omega_1,\omega_2$ are comparable,

	we can apply
  Proposition~\ref{prop:clear}.  Thus, assume $\omega_1,\omega_2$ are
  incomparable and let $d_1=|\omega_1\setminus \omega_2|$ and
  $d_2=|\omega_2\setminus \omega_1|$. W.l.o.g.\ assume $d_1\leq
  d_2$. Also note that $d_1 > 0$.  We use $K'$ with
  $\mod(K')=\{\omega_1^+,\omega_1\cup \omega_2\}$ where $\omega_1^+$
  adds $d_1$ elements from $\omega_2\setminus \omega_1$ to $\omega_1$.
  Thus, $\omega_1^+\subseteq \omega_1\cup \omega_2$ and we can choose
  $K'$ from $\L_{\mathit{Horn}}$.  Moreover, let $\mu\in\L_{\mathit{Horn}}$ such that
  $\mod(\mu)=\{\omega_1,\omega_2,\omega_1\cap \omega_2\}$.  We have
  the following distances (note that $d(\omega_2,\omega_1^+)=d_1+(d_2-d_1)$).
$$
\begin{array}{lccc}
           &     \omega_1^+      &        \omega_1\cup \omega_2 &    K'\\
\omega_1          &     d_1      &        d_2      &    d_1 \\
\omega_2          &     d_2  &         d_1     &     d_1\\
\omega_1\cap \omega_2    &     2d_1    &        d_1+d_2    &    > d_1\\
\end{array}
$$
Hence, $\Delta^{H}_\mu(K') \equiv K$ as desired.
\end{proof}

Our final result concerns distributability in the $\mathit{Horn}$ fragment.
We show that some KBs with three models can be distributed.

\begin{proposition}
Let $K$ be a KB such that $\mod(K) = \{\omega_1,\omega_2,\omega_3\}$. If $\omega_1$, $\omega_2$ and $\omega_3$ are not all pairwise incomparable, then $K$ is $\mathit{Horn}$-distributable w.r.t. $\Delta^{H,\otimes}$ with $\otimes \in \{\sum,\GMAX,\GMIN\}$.
\end{proposition}

\begin{proof}
  If $K$ is $\mathit{Horn}$-expressible, then the result follows from Proposition~\ref{prop:clear}.
  If $K$ is not $\mathit{Horn}$-expressible, then we do a case analysis on the number of pairwise incomparable models of $K$.

	\textit{Case 1.}
	If exactly one pair of models of $K$ are incomparable, then we can assume without loss of generality that it is $\omega_1$ and $\omega_2$.
	It follows then that $\omega_3\neq\omega_1\cap\omega_2$.
	Also, there must be distinct atoms $a$ and $b$ such that $a\in\omega_1$, $a\notin\omega_2$ and $b\in\omega_2$, $b\notin\omega_1$.
	We consider a constraint $\mu\in \L_{\mathit{Horn}}$ such that $\mod(\mu)=\{\omega_1, \omega_2, \omega_3, \omega_1\cap \omega_2\}$.

	\textit{Case 1.1.} If $\omega_1 \subseteq w_3$ and $w_2\subseteq \omega_3$, then we take a globally new atom $c$ and KBs $K_1$ and $K_2$ such that:
	\begin{itemize}
	  \item $\mod(K_1)= \{\omega_1\cup \{b\},\omega_2 \cup \{a\},\omega_3, (\omega_1\cap\omega_2)\cup\{a,b\}\}$
	  \item $\mod(K_2)= \{\omega_1,\omega_2, \omega_3\cup\{c\},\omega_1\cap \omega_2\}$
	\end{itemize}
	It is easy to see that $K_1$ and $K_2$ are $\mathit{Horn}$-expressible.
	Considering, now, the profile $E=(K_1,K_2)$, we obtain the following distances:
		$$
		\begin{array}{llllll}
		                        &	K_1	&		K_2 & \sum & \GMAX  & \GMIN\\
		\omega_1        	&	1	&		0   &  1 & (1,0) & (0,1)\\
		\omega_2        	&	1	&		0   &  1 & (1,0) & (0,1)\\
		\omega_3	        &	0	&		1   &   1 & (1,0) & (0,1)\\
		\omega_1\cap \omega_2	&	 2	&	0    & 2    & (2,0) & (0,2)\\
		\end{array}
		$$
	So for each $\oplus \in \{\sum,GMax,GMin\}$ we obtain that
	$\Delta_\mu^{H,\oplus}(E) \equiv K$.

	\textit{Case 1.2.} If $\omega_3 \subseteq \omega_1$ and $\omega_3 \subseteq \omega_2$, then $\omega_3 \subseteq \omega_1 \cap \omega_2$.
	Moreover, since $\omega_3\neq\omega_1\cap\omega_2$, it actually holds that $\omega_3 \subset \omega_1 \cap \omega_2$.
	Thus there exists an atom $c$ such that $c \in (\omega_1 \cap \omega_2)$ and $c \notin \omega_3$.
	We now take KBs $K_1$, $K_2$, $K_3$ and $K_4$ such that:
	\begin{itemize}
	  \item $\mod(K_1)= \{\omega_1\cup \{b\},\omega_2 \cup \{a\},\omega_3, (\omega_1\cap \omega_2)\backslash\{c\}, (\omega_1\cap\omega_2)\cup\{a,b\}\}$
	  \item $\mod(K_2)= \{\omega_1,\omega_2 \cup\{a\}, \omega_3, (\omega_1\cap \omega_2)\cup \{a\}\}$
	  \item $\mod(K_3)= \{\omega_1\cup\{b\},\omega_2, \omega_3,(\omega_1\cap \omega_2)\cup\{b\}\}$
	  \item $\mod(K_4)= \{\omega_1,\omega_2, \omega_3\cup\{c\},\omega_1\cap \omega_2\}$
	\end{itemize}
	It is easy to see that $K_1$, $K_2$, $K_3$ and $K_4$ are $\mathit{Horn}$-expressible.
	Considering, now, the profile $E=(K_1,K_2,K_3,K_4,K_4)$, we obtain the following distances:

		\begin{table}[h]
		\centering
		\scalebox{0.82}{
		\begin{tabular}{lllllllll}
											                        & $K_1$ & $K_2$ & $K_3$ & $K_4$ & $K_4$ & $\sum$ & $\GMAX$  & $\GMIN$\\
		$\omega_1$        							 &	1	  &	   0    &   1    &     0   &     0    &     2       & (1,1,0,0,0) & (0,0,0,1,1)\\
		$\omega_2$        							 &	1	  &	   1    &   0    &     0   &     0   &      2       & (1,1,0,0,0) & (0,0,0,1,1)\\
		$\omega_3$	        						 &	0	  &	   0    &   0    &     1    &    1    &     2       &  (1,1,0,0,0)  &(0,0,0,1,1)\\
		$\omega_1\cap \omega_2$	&	1 	 &    1     &  1      &   0     &	0    &     3       & (1,1,1,0,0) & (0,0,1,1,1)\\
		\end{tabular}
		}
		\end{table}

	So for each $\oplus \in \{\sum,GMax,GMin\}$ we obtain that
	$\Delta_\mu^{H,\oplus}(E) \equiv K$.

	\textit{Case 2.}
	If exactly two pairs of models of $K$ are incomparable, then we can assume without loss of generality that it is $w_1, w_2$ and $w_2, w_3$.
	We consider a constraint $\mu\in \L_{\mathit{Horn}}$ such that $\mod(\mu)=\{\omega_1, \omega_2, \omega_3, \omega_1\cap \omega_2, \omega_2\cap\omega_3\}$.
	Then there must be distinct atoms $a$ and $b$ such that $a\in \omega_1$, $a\notin\omega_2$ and $b\in\omega_2$, $b\notin\omega_1$.
	Further, there must be distinct atoms $c$ and $d$ such that $c\in \omega_2$, $c\notin\omega_3$ and $d\in\omega_3$, $d\notin\omega_2$.

	\textit{Case 2.1.} If $w_1\subseteq w_3$, then we get that $c\notin\omega_1$ and $a\in\omega_3$.
	We take KBs $K_1$ and $K_2$ such that:
	\begin{itemize}
	  \item $\mod(K_1)= \{\omega_1\cup \{c\},\omega_2,\omega_3 \cup\{c\}, (\omega_1\cap\omega_2)\cup\{c\},(\omega_2\cap\omega_3) \cup \{c\}\}$
	  \item $\mod(K_2)= \{\omega_1,\omega_2\cup \{a\},\omega_3, (\omega_1\cap\omega_2)\cup\{a\}, (\omega_2\cap\omega_3)\cup\{a\}\}$
	\end{itemize}
	It is easy to see that $K_1$ and $K_2$ are $\mathit{Horn}$-expressible.
	Considering, now, the profile $E=(K_1,K_2)$ and keeping in mind that $c\notin\omega_1$ and $a\in\omega_3$, we obtain the following distances:
		$$
		\begin{array}{llllll}
		                        &	K_1	&		K_2 &  \sum & \GMAX  & \GMIN\\
		\omega_1        	&	1	&		0   &  1 & (1,0) & (0,1)\\
		\omega_2        	&	0	&		1   &  1 & (1,0) & (0,1)\\
		\omega_3	        &	1	&		0   &   1 & (1,0) & (0,1)\\
		\omega_1\cap \omega_2	&	1	&	1 &	 2    & (1,1) & (1,1)\\
		\omega_2\cap \omega_3	&	1	&	1 &	 2    & (1,1) & (1,1)\\
		\end{array}
		$$
	So for each $\oplus \in \{\sum,GMax,GMin\}$ we obtain that
	$\Delta_\mu^{H,\oplus}(E) \equiv K$.

	\textit{Case 2.2.} If $w_3\subseteq w_1$, then we get that $b\notin\omega_3$ and $d\in\omega_1$.
	We take KBs $K_1$, $K_2$ and such that:
	\begin{itemize}
	  \item $\mod(K_1)= \{\omega_1\cup \{b\},\omega_2,\omega_3 \cup\{b\}, (\omega_1\cap\omega_2)\cup\{b\},(\omega_2\cap\omega_3) \cup \{b\}\}$
	  \item $\mod(K_2)= \{\omega_1,\omega_2\cup \{d\},\omega_3, (\omega_1\cap\omega_2)\cup\{d\}, (\omega_2\cap\omega_3)\cup\{d\}\}$
	\end{itemize}
	It is easy to see that $K_1$ and $K_2$ are $\mathit{Horn}$-expressible.
	Considering, now, the profile $E=(K_1,K_2)$ and keeping in mind that $b\notin\omega_3$ and $d\in\omega_1$, we obtain the following distances:
		$$
		\begin{array}{llllll}
		                        &	K_1	&		K_2 &  \sum & \GMAX  & \GMIN\\
		\omega_1        	&	1	&		0   &  1 & (1,0) & (0,1)\\
		\omega_2        	&	0	&		1   &  1 & (1,0) & (0,1)\\
		\omega_3	        &	1	&		0   &   1 & (1,0) & (0,1)\\
		\omega_1\cap \omega_2	&	1	&	1 &	 2    & (1,1) & (1,1)\\
		\omega_2\cap \omega_3	&	1	&	1 &	 2    & (1,1) & (1,1)\\
		\end{array}
		$$
	So for each $\oplus \in \{\sum,GMax,GMin\}$ we obtain that
	$\Delta_\mu^{H,\oplus}(E) \equiv K$.
	The cases when $\omega_2\subseteq \omega_3$ or $\omega_3\subseteq \omega_2$ are symmetric.
This concludes our case analysis, as any other remaining case results in either all of the interpretations $\omega_1$, $\omega_2$ and $\omega_3$ being pairwise incomparable, or in $K$ being $\mathit{Horn}$-expressible.
\end{proof}

The remaining case (i.e., $\mod(K)=\{\omega_1, \omega_2,\omega_3\}$ with $\omega_1$, $\omega_2$, $\omega_3$ pairwise incomparable), as well as the more general case when $K$ has an arbitrary number of models is subject to ongoing work.

\section{Conclusion}
In this paper we have proposed the notion of distributability and we have studied the properties of several merging operators with respect to different fragments of propositional logic. Our results are summarized in Table~\ref{table:summary}.
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|}
  \hline
  & $\mathit{1CNF}$ & $\mathit{2CNF}$ & $\mathit{Horn}$ \\ \hline
  simplifiable w.r.t.\ $\Delta^D$ & $\times$ & $\times$ & $\times$ \\ \hline
  simplifiable w.r.t.\ $\Delta^H$ & $\times$ & $\checkmark$ & $\circ$ \\ \hline
  distributable w.r.t.\ $\Delta^{D,\sum}$ & $\checkmark$ & $\checkmark$ & $\checkmark$ \\ \hline
  distributable w.r.t.\ $\Delta^{H,\sum}$ & $\times$ & $\checkmark$ & $-$ \\ \hline
  distributable w.r.t.\ $\Delta^{H,GMax}$  & $-$ & $\checkmark$ & $-$ \\ \hline
  distributable w.r.t.\ $\Delta^{H,GMin}$ & $-$ & $\checkmark$ & $-$ \\ \hline
\end{tabular}
\caption{Summary of Results \label{table:summary}}
\end{table}
The symbol $\times$ means that only ``trivial'' knowledge bases (belonging to the considered fragment) can be distributed with the corresponding operator.
Alternately, $\checkmark$ means that any knowledge base can be distributed.
Symbol $-$ means we know that some non-trivial knowledge bases can be distributed, and finally $\circ$ means that some, but not all, non-trivial bases can be simplified.
Interestingly, the picture emerging from Table~\ref{table:summary} is that merging operators behave quite differently depending on the distance and aggregation function employed, in a way that does not lend itself to simple categorization.
For instance, our results on simplifiability imply that using Dalal revision to $\L_{\mathit{1CNF}}$ KBs never takes us outside the $\mathit{1CNF}$ fragment; applying the same revision operator to $\L_{\mathit{Krom}}$ KBs can produce any KB in $\L$; and applying it to $\L_{\mathit{Horn}}$ KBs can produce some, though not all possible KBs.

Several questions are still open for future work. We plan to study the exact characterization of what can (and cannot) be distributed, in order to replace the symbols $-$ and $\circ$ in the previous table. Other merging operators
can also be integrated to our study. Some of our results on distributability require the addition of new atoms to the interpretations. We want to  determine whether similar results can be obtained without modifying the set of propositional variables, in particular for the $\mathit{2CNF}$ fragment.
We are also interested in the number of knowledge bases needed to
distribute knowledge: given an integer $n$, a knowledge base $K$ and a
merging operator $\Delta$, is it possible to distribute $K$
w.r.t. $\Delta$ such that the resulting profile contains at most $n$
knowledge bases?
This paper was a first step to understand the limits of distributability; the actual construction of the profile and complexity of this process are important questions that will be tackled in future research.
Finally, we also consider applying the concept of distributability to
non-classical formalisms, in particular in connection with merging
operators proposed for logic programs  \cite{DelgrandeSTW13}.

\section*{ Acknowledgments}
This work was supported by the Austrian Science Fund (FWF)
under grant
P25521.

redpajama_set_name: RedPajamaArXiv


Gavin, as amember of the League's advocacy team, implements targeted campaigns and programs to help protect, restore and advocate for the health of the Lake Tahoe Basin. He conducts environmental analysis and research, and assists with federal, state and local advocacy. Gavin is responsible for the development, implementation, and enforcement of applicable planning policy within the greater Lake Tahoe region, with an emphasis on land use and transportation planning.
Gavin brings a decade of experience in environmental science, policy, planning, and advocacy with nonprofit organizations. Prior to joining the League's staff, he managed monitoring and restoration programs for the Alpine Watershed Group, worked for the Sierra Nevada Alliance providing support to conservation groups across the Sierra Nevada to implement strong natural resource plans and policy, and advocated for federal climate change policy in Washington, D.C. for the Pew Charitable Trusts. Gavin obtained his Master of Environmental Science and Management from the Bren School at the University of California, Santa Barbara, and his B.S. in Geology from the University of Washington. He joined the League's staff in 2018.
When not working, Gavin spends much of his time advocating for bicycle and pedestrian accessibility and safety and enjoying all of the outdoor recreation opportunities the Sierra has to offer.