Uso della logica fuzzy all'interno di ontologie scritte in OWL2, basato sul lavoro di U. Straccia e F. Bobillo. Presentazione fatta per il corso di Web Semantico @ Università degli studi di salerno.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Uso della logica fuzzy all'interno di ontologie scritte in OWL2, basato sul lavoro di U. Straccia e F. Bobillo. Presentazione fatta per il corso di Web Semantico @ Università degli studi di salerno.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
2. 1) Introduzione - Non
trattabilità di IA
IA non è trattabile:
● Verifica della consistenza e calcolo della rete minima
sono problemi NP-completi
Sottoclassi trattabili di IA:
● SAc e SA, esprimibili con Point Algebra
Costo computazionale contenuto
VS
Potere espressivo limitato
3. fuz
Introduzione - IA
Creare un'estenzione fuzzy di IA:
● Ad ogni relazione atomica tra intervalli temporali viene
associato un grado di preferenza in [0,1]
● E' possibile così indicare il grado di necessità di
soddisfacibilità di una relazione
IAfuz è una generalizzazione di IA: stesso potere espressivo
e stessa intrattabilità:
● Come per IA, è necessario trovare sub-algebra trattabile
di IAfuz
4. fuz
Introduzione - IA (2)
● Si dimostrerà che PAc, PA, SAc e SA possono essere
estese in modalità fuzzy.
● Si dimostrerà che le estensioni fuzzy di SAc e SA sono
sub-algebre trattabili di IAfuz
5. 2) Interval Algebra
In IA la conoscenza temporale è rappresentabile con un
grafo:
● nodi rappresentano intervalli
● archi etichettati con le relazioni ugualmente possibili tra
i 2 intervalli (13 relazioni possibili)
I1 (rel1, rel2, ...) I2
6. Interval Algebra (2)
Singleton labelling di una rete IA N: assegnazione di una
relazione atomica ad ognuno dei suoi archi.
Soluzione di N: singleton labelling che soddisfa tutti i vincoli
della rete ed è consistente (le relazioni tengono)
SOL(N): insieme di soluzioni
7. Estensione fuzzy di IA
Nel nostro approccio le relazioni tra due intervalli sono nella
forma:
I1 (rel1[α1], rel2[α2], ...) I2
αi : grado di preferenza della relazione i-esima
nell'intervallo [0,1]
degR(reli) = grado di preferenza assegnato dalla relazione
R alla relazione atomica di Allen reli
8. Estensione fuzzy di IA (2)
Abbiamo esteso IA ad una nuova algebra fuzzy IAfuz
definita dall'insieme:
I = {b[α1], a[α2], m[α3], mi[α4], d[α5], di[α6], o[α7], oi[α8], s
[α9], si[α10], f[α11], fi[α12], eq[α13]}
con αi ∈ [0,1], αi ∈ R, i = 1,..., 13
Ponendo invece αi ∈ {0,1} ri-otteniamo il framework classico
9. fuz
Chiusura di IA rispetto a...
● Data una relazione R, l'operatore di inversione R-1 è
definito come
R-1 = (rel1-1[α1],...,rel13-1[α13])
dove reli-1 è definito nella tabella di Allen
● Date due relazioni R' e R'', la congiunzione R = R' x R'' è
definita come:
R = (rel1[α1],...,rel13[α13])
dove αi = min{α'i,α''i} i ∈ {1,...,13}
10. fuz
Chiusura di IA rispetto a...
(2)
● La disgiunzione R = R' + R'':
R = (rel1[α1],...,rel13[α13])
dove αi = max{α'i,α''i} i ∈ {1,...,13}
● Composizione R = R' ° R'':
R = (rel1[α1],...,rel13[α13])
dove αi = maxj,k: reli ∈ {relj ° relk}min{α'j,α'k}
i,j,k ∈ {1,...,13}
11. Consistenza
Nella IA classica il singleton labelling parziale è localmente
consistente se soddisfa tutti i vincoli
In IAfuz: consistenza locale graduata
Grado di un labelling singleton parziale s:
● Se l'assegnamento è inconsistente: degN(s)=0
● altrimenti degN(s) uguale al grado di preferenza del
vincolo meno soddisfacente
12. Consistenza (2)
Una rete IAfuz è k-consistente se per ogni insieme di k-1
nodi, ogni assegnamento con un grado di consistenza
locale α, è estensibile a qualsiasi altra k-esima variabile
mantenendo lo stesso grado α.
Path consistency: k-consistency con k=3
13. Calcolo della rete minima
equivalente
Due reti N1 e N2 sono equivalenti se coinvolgono le stesse
variabili e per ogni singleton labelling completo s, degN1(s)
= degN2(s)
La rete minima è quella "più esplicita" tra tutte le reti
equivalenti
Una rete IAfuz N è minima sse per ogni relazione Rij tra due
intervalli (Ii,Ij) e per ogni relazione relk[α] appartenente a Rij
c'è un singleton labelling s di N che assegna relk a Rij e tale
che degN(s) = α
14. fuz
3) Da PA a PA
La point algebra classica PA si basa sulla nozione di punti
temporali:
● Tre relazioni di base: <, = e >
● Possibilità di esprimere 8 diverse relazioni: Ø, <, ≤, =,
>, ≥, ≠, ?
Un particolare subset di PA è la PAc (PA senza ≠), che ha
interessanti poprietà computazionali:
L'algoritmo di path-consistency ottiene la minimalità in
tempo polinomiale [O(n3)]
15. fuz
Da PA a PA (2)
Definiamo PAfuz considerando punti invece di intervalli e
relazioni PA invece di relazioni di Allen
● PAfuz è definito sull'insieme
I = {< [α1], = [α2], > [α3]} dove αi ∈ [0,1], αi ∈ R, i = 1, 2,
3
● PAcfuz è la subalgebra di PAfuz definita sull'insieme
I = {< [α1], = [α2], > [α3]} dove αi ∈ [0,1], αi ∈ R, i = 1, 2,
3
α2≥ min {α1, α3}
16. fuz
Da PA a PA (3)
L'idea è di eliminare il corrispettivo fuzzy di ≠, che
corrisponde alla classe di relazioni PAfuz (< [α1], = [α2], >
[α3]) tali che α2 < α1 e α2 < α3
Formalizziamo la relazione tra PAfuz e PAcfuz
Data una relazione PAfuz (IAfuz) Rfuz, il suo α-taglio Rα è la
relazione PA (IA) fatta di relazioni atomiche reli tali che
degRfuz(reli) ≥ α
Proposizione 1: Data una relazione R ∈ PAfuz, R ∈ PAcfuz
sse ∀α ∈ [0, 1] Rα ∈ PAc
17. fuz
Da PA a PA (4)
● PAcfuz è un algebra
● La prova di completezza dell'algoritmo di Path-
Consistency di PAc può facilmente essere estesa a
PAcfuz
Quindi può essere provato che l'applicazione dell'algoritmo
di path consistency a Pacfuz trova la rete minima
equivalente
18. Estenzione fuzzy
dell'algebra dei punti
I risultati sulla tracciabilità delle point algebras valgono
anche nel contesto dell'interval algebra
Sono state definite due sub-algebras di IA, dette SAc e SA
relazionate con PAc e PA
Le relazioni SA e SAc sono quelle relazioni di IA esprimibili
dalle relazioni di PA e PAc
19. Estenzione fuzzy
dell'algebra dei punti (2)
Tutte le proprietà computazionali di PA e PAc sono
mantenute:
● SAc path consistent sono minimali
● minimalità di 4-subnetworks assicura minimalità di SA
networks
Generalizziamo le definizioni di SAc e SA al framework
fuzzy, studiando le relazioni tra le algebre classiche e le
estenzioni fuzzy, chiamate SAcfuz e SAfuz
20. Da relazione tra intervalli a
fuz
rete PA
Possiamo tradurre una relazione IAfuz tra due intervalli in
una rete PAfuz che interessa i 4 endpoints:
Definizione: Consideriamo la classe delle reti PAfuz con 4
nodi, denotati con {I1-,I1+,I2-,I2+} , ed indichiamo le loro
relazioni binare come R12--, R12-+, R12+-, R12++, R11-+, R22-+.
SPAfuz è la classe di reti tali che R11-+ = {< [α1]} e R22-+ = {<
[α2]}, dove α1, α2 ∈ [0,1]
21. Da relazione tra intervalli a
fuz
rete PA (2)
Una rete N ∈ SPAfuz verrà denotata dalla 6-tupla delle sue
relazioni PAfuz.
N ha al più 13 distinte soluzioni con grado di
soddisfacimento > 0, che corrispondono alle possibili
disposizioni dei due intervalli IA.
Chiamiamo questo set SOLIA ⊆ {<,=,>}6
Ogni elemento s ∈ SOLIA corrisponde ad una relazione
atomica di IA
Chiamiamo fIA la funzione da SOLIA a IA
22. fuz
La sub-algebra SA
Definizione 3: Sia R una relazione R ∈ IAfuz; R ∈ SAfuz sse
∃ N minima ∈ SPAfuz tale che ∀ s ∈ SOLIA degN(s) = degR
(fIA(s)). Denotiamo questa rete N con C(R).
Ovvero, le relazioni SAfuz sono quelle che possono essere
tradotte in reti PAfuz preservando lo stesso insieme di
soluzioni cosi come lo stesso grado di soddisfacimento
La subalgebra SAcfuz ⊆ SAfuz ⊆ IAfuz è composta dalle
relazioni che possono essere espresse con reti PAcfuz:
Definizione 4: Sia R una relazione R ∈ SAfuz; R ∈ SAcfuz
sse C(R) è una rete PAcfuz.
23. fuz
La sub-algebra SA (2)
Come per l'Interval Algebra classica, IAfuz è più espressiva
della Point Algebra: solo un frammento di IAfuz può essere
tradotto in PAfuz.
SAfuz e SAcfuz sono state definite tramite una
generalizzazione di SA e SAc: mantengono la stessa
relazione presente tra PA e PAfuz e PAc e PAcfuz.
Proposizione 2: Data una relazione IAfuz R, R ∈ SAfuz sse
∀α ∈ [0,1] Rα ∈ SA.
Proposizione 3: La proposizione 2 vale anche per SAcfuz
24. fuz
La sub-algebra SA (3)
Proposizione 4: SAfuz (SAcfuz) è un'algebra rispetto alle
operazioni di inversione, congiunzione e composizione
● Inversione: se I1 R I2, allora I2 R-1 I1
● Congiunzione e composizione:
○ Se * ∈ {⊗,◦} allora R = R1 * R2 sse ∀α ∈ [0, 1] Rα =
R1α* R2α
○ Se R1 e R2 sono relazioni SAfuz (SAcfuz) per la Prop. 2
(Prop. 3) R1α e R2α sono relazioni SA (SAc)
○ SA (SAc) è un'algebra quindi Rα ∈ SA (Rα∈ SAc)
...
25. fuz
La sub-algebra SA (4)
○ Tutti gli α-tagli di R appartengono a SA (SAc) e per
Prop. 2 (Prop. 3) R ∈ SAfuz (R ∈ SAcfuz)
Possiamo ora provare che le proprietà computazionali
rilevanti sono mantenute nel framework fuzzy
26. 5 - Due subalgebre trattabili
fuz
di IA
Dimostreremo che SAcfuz e SAfuz sono classi trattabili di
IAfuz, (mantengono le poprietà computazionali di SAc e SA)
Definizione 5: Data una rete IAfuz (PAfuz) N, il suo α-taglio
Nα è la rete IA (PA) con gli stessi nodi di N, ma i cui vertici
sono etichettati con gli α-tagli degli vertici corrispondenti di
N
Proposizione 5: Una rete IAfuz (PAfuz) N è minima sse ∀α
∈ [0,1] Nα è minima
Proposizione 6: Una rete IAfuz (PAfuz) N è k-consistente sse
∀α ∈ [0,1] Nα è k-consistente
27. Proprietà computazionali
Proposizione 7: Se una rete SAcfuz è path-consistent, allora
è anche minima
Prova: Sia N una rete SAcfuz path-consistent. Per le
proposizioni 3 e 5, ∀ α ∈ [0,1], Nα è una rete SAc path-
consistent.
Poichè le reti SAc path-consistent sono minime, tutti gli α-
tagli di N sono minimi, e quindi lo è anche N (per la
proposizione 5)
28. Proprietà computazionali
Proposizione 8: Sia N una rete SAfuz. Se tutte le sue 4-
subnetworks sono minime, allora N è minima
Prova:
Per la proposizione 5, ∀ α ∈ [0,1], tutte le 4-subnetworks
di Nα sono anch'esse minime.
Per la proposizione 2, abbiamo che ∀ α ∈ [0,1], Nα è una
rete SA
Tenendo conto dei risultati per le reti SA classiche, tutti gli
α-tagli di N sono minimi
Per la proposizione 5, anche N è minima
29. Algoritmi
Queste proposizioni possono essere utilizzate per
sviluppare un algoritmo polinomiale che calcola la rete
minima equivalente ad una rete SAcfuz o SAfuz N.
Per reti SAcfuz è sufficiente utilizzare l'algoritmo di path
consistency con complessità O(k*n3)
Per le reti SAfuz si può utilizzare una generalizzazione dei
AAC, con complessità O(k*n4)
30. Conclusioni
Diversamente da altri lavori della letteratura questo si
focalizza sull'aspetto qualitativo della rappresentazione
Si è dimostrato come le proprietà computazionali delle
subalgebre trattabili si possano estendere a IAfuz
Il prossimo passo è l'integrazione di questo framework
qualitativo in uno quantitativo