Relativized limitations of left set technique and closure classes of sparse sets

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A number of theorems are proved by introducing the notion of k -families of sets of strings, and an algorithm which outputs the sets of certain k-families is given. The algorithm is used to disjunctively reduce the left set (or 1wdsr set) to a sparse
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  Values, Decisions, and Inner Peaceor Who Am I and Where am I Going?  Nothing can bring you peace but yourself. Nothing can bring you peace but the triumph of principles.  !alph Waldo "merson #a$e you e$er stopped to really thin% about &ho you are and &here you are going in your life? It is not something &e li%e to d&ell on $ery much. 'he ans&ers aren(t al&ays so clear or easy to ma%e in this sometimes frightening &orld. We might ha$e a good idea of some aspects of our life, li%e &hat &e &ant to do academically or professionally. )ut many parts of our li$es remain undisco$ered. 'his may be due to fear of finding &hat might be there, or it may be because &e simply don(t %no& ho& to loo%. *e$eral of our great thin%ers had these &ise &ords regarding ho& &e become &ho &e are. 'hin% about these +uotes for a moment- 'he mind is e$erything &hat you thin%, you become.  )uddha- /ou are today &here your thoughts ha$e brought you. /ou &ill be tomorro& &here your thoughts ta%e you.0 1ames Allen- We become &hat &e thin% about.  Napoleon #ill- A man is &hat he thin%s about all day long.  !alph Waldo "merson- As a man 'hin%eth in his heart, so is he  Da$id, the psalmist 'hese men understood that &hat happens in our regular thin%ing patterns brings about the type of  person each of us becomes. *ometimes it is difficult to ma%e the best decisions. We find oursel$es struggling to choose bet&een se$eral alternati$es. )ut it is our decisions that &e ma%e on a moment to moment basis create the future that &e &ill li$e. All decision ma%ing comes do&n to ho& &e $alue those things on &hich &e are deciding  al&ays. When you %no& &hat is most important to you, ma%ing decisions is a simple process. When you aren(t sure &hat you $alue most in a situation, ma%ing the best one for you is more difficult.'hin% about the people &ho &e tend to respect the most in our culture. 'hey are usually those &ho ha$e clearly defined $alues and li$e by them. 2ahatma Gandhi &as a perfect e3ample of a person &ho &as $ery clear about &hat &as most important to him. Despite going up against impossible odds, li$ing according to his highest $alues ultimately brought about the freeing of an entire nation. Gandhi &as $ery clear about his $alues. #e %ne& that his choices and beha$ior follo&ed them. #e &as dri$en by his $alues instead of being dri$en by his emotions or the circumstances in his en$ironment.Why should &e disco$er our Values?'here is a principle in )uddhism called Dharma. It is a rather comple3 principle, but one aspect of it has to do &ith the idea of a 4igsa& pu55le. 6onsider the possibility that each one of us that has e$er li$ed is a specific piece of an enormous pu55le of se$eral billion pieces 7one for each person8. 'his is a $ery large pu55le. 6onsider, further, that your o&n personal piece of this gigantic pu55le is a specific si5e and shape and fits correctly in only one precise place in this pu55le. /our piece of this pu55le does not fit in any other place on the   pu55le board. In other &ords, you are not able to be another pu55le piece you can only be your o&n. Dharma teaches us that &hen you find out &hat your pu55le piece is all about, you find satisfaction in life you feel fulfilled, happy, content, and &orth&hile. When a person aimlessly &anders about not e$en %no&ing about the  pieces of the pu55le, or tries to be someone else(s piece, thin%ing that is the appropriate &ay to be and do things, this person is li%ely to find confusion, unhappiness, despair. As 'horeau said, 9'he mass of men lead li$es of +uiet desperation.: 'his is &hat he &as tal%ing about. People are not li$ing according to their o&n pu55le piece, &hich is uni+uely theirs and entirely necessary to find out. 'his is stressful.What happens &hen &e do li$e according to our o&n pu55le piece? 'he natural conse+uences are inner  peace, &isdom and happiness. We feel fulfilled and satisfied &ith oursel$es and the direction in &hich our life is going &hen &e are certain about &ho &e are. What happens &hen &e don(t disco$er our real purpose for being? What are the conse+uences of not %no&ing &hat our inner nature is and follo&ing that path but instead, follo&ing someone else(s perfect &ay &hich is not perfect for us? Anthony !obbins describes &hat commonly happens &hen people don(t ta%e a good loo% at &ho they are and &here they are going. #e calls it the Niagara *yndrome and it goes li%e this 9;<ife is li%e a ri$er, and most people 4ump on the ri$er of life &ithout e$er really deciding &here they &ant to end up. *o, in a short period of time, they get caught up in the current current e$ents, current fears, and current challenges. When they come to for%s in the ri$er, they don=t consciously decide &here they &ant to go, or &hich direction is right for them. 'hey merely >go &ith the flo&.> 'hey become a  part of the mass of people &ho are directed by the en$ironment instead of by their o&n $alues. As a result, they feel out of control. 'hey remain in this unconscious state until one day the sound of the raging &ater a&a%ens them and they disco$er that they are  feet from Niagara @alls in a boat &ith no oars. At this point, all they can say is, >oh, shoot> )ut by then it=s too late. 'hey are going to ta%e a fall. *ometimes it=s an emotional fall. *ometimes it=s a physical fall. *ometimes it=s a financial fall. It is li%ely that &hat e$er challenges you ha$e in your life currently could ha$e been a$oided by some better decisions upstream.: 7!obbins, BC0B8. A similar analogy is about the person &ho spends his entire life climbing up the ladder of success only to reali5e, &hen he arri$es at the top of the ladder, that his ladder &hich he has been climbing for such a long time, is leaning against the &rong &all. We commonly call the e3perience this person has &hen he comes to this reali5ation of leaning against the &rong &all, or coming upon the &aterfall, a midlife crisis. 'his person goes through deep emotional trauma as%ing himself such penetrating +uestions as, 9#o& did I get to this point in my life?> Er, >What ha$e I done &ith my life?> As 'ony !obbins mentions, ma%ing the choice to flo& in a different ri$er or climb up the different ladder &ould ha$e resulted in a far different e3perience.Fno&ing &hat our $alues are and then learning to li$e by them is one of the most po&erful &ays to gain inner peace and decrease stress le$els. Not only does this apply to the bigger life decisions, but our e$eryday choice ma%ing as &ell. As an e3ample, consider the man &ho deeply $alues his relationship &ith his &ife and %ids. #e is a family man. )ut fre+uently, during his &or%day, he spends e3cessi$e amounts of time &ith other &omen, ta%ing them out to lunch and buying e3pensi$e gifts for them. If this man has a conscience of any %ind, he &ill $ery li%ely feel +uite unpleasant feelings &ithin himself because his actions do not match the things he  considers that are most important. Another person &ho $alues honesty highly &ill feel some $ery uncomfortable feelings &hen she cheats on a test or shoplifts a nice shirt that she really &ants. When our actions are not in line &ith our $alues, the natural emotional conse+uence is stress.En the other hand, the person &ho $alues the personality trait of lo$e and compassion, and &ho spends large amounts of time $olunteering in a hospice &ill find deep feelings of peace and contentment because her actions are in line &ith her beha$ior. 'he more our beha$ior is out of line &ith our $alues, the more stress and inner chaos &e &ill feel.'his chapter is about helping us find out &hat is our o&n pu55le piece and then disco$ering &hat this  piece of the pu55le is all about. It is a chapter on real self0disco$ery. )ecause once &e find out &ho &e really are and &hat our life is about, &e can then ma%e choices that support that life instead of a life &here &e are &andering aimlessly li%e a boat &ith no rudder to guide it along. Author(s anecdoteEn one occasion, &hen I &as about B, I &as 4ogging. I &as nearing the end of a fairly long run. I &as feeling $ery good as the endorphins &ere cruising and second &ind &as &ell in place. I &as at that place &here I felt li%e I could 4og fore$er. I &asn=t really thin%ing of anything in particular &hen suddenly an o$er&helming thought occurred to me. It sounded something li%e this, >Damn 'his is my life I am li$ing here 2y life is nobody else(s to li$e. I can only li$e this life and all I &ill e$er ha$e is my life. )ut &hen I go along &ith the cro&d, I=m not li$ing my life. When I follo& the direction that my parents, my teachers, my coaches thin% is best for me, I am not li$ing my o&n life. 2y life is mine to choose. And if I don=t start choosing, it=s going to pass me by.: I %ne& then and there that I did not &ant to come to the end of my life and thin% that I had settled for mediocrity that I had gotten so caught up in the day0to0day stuff that I had lost all a&areness of &hat &as really important to me. I did not &ant my final &ords, &hen it &as time to depart from this life, to be, >if only.> I &arn you. <oo% at e$ery path closely and deliberately. 'ry it as many times as you thin% necessary. 'his +uestion is one that only a $ery old man as%s ; Does this path ha$e a heart? If it does, the path is good if it doesn(t, it is of no use.  6arlos 6astaneda@inding out &hat(s most Important  Eur ValuesIn order for us to ma%e positi$e change in the direction of our o&n true path, there are se$eral beliefs that &e must firmly maintain in our minds that &ill support us as &e begin our 4ourney. #ere are these beliefsC. We must first belie$e that &e are capable of changing no&. !egardless of our current situation &e ha$e the capacity and the ability to ma%e any changes that &e feel are appropriate.. We must also ha$e the belief that if &e are going to create long0term change in our li$es, that &e are responsible. Nobody else is going to do it for us. It re+uires our o&n decision, our o&n moti$ation, and our o&n action.. We must ha$e the belief that if &e set our sites in a ne& direction, and then mo$e confidently in that direction, &e &ill successfully arri$e near the place &e &anted to go. #enry Da$id 'horeau &rote 9I learned this, at least, by my e3periment that if one ad$ances confidently in the direction of his dreams, and endea$ors to li$e the life &hich he has imagined, he &ill meet &ith success in uncommon hours.:B. We must be certain that our $alues determine our actions and beha$iors. We may not be clear about &hat &e $alue, but our choices are dependent on &hat &e feel is most important to us. In other  &ords, all decision ma%ing is based on $alues clarification. It is appropriate to as%, at this point, &hat a $alue really is. !o%each said that $alues are 9enduring  beliefs that a specific mode of conduct or end0state of e3istence is personally or socially preferable to an opposite or con$erse mode of conduct or end0state of e3istence. Allport described a $alue as 9a belief upon &hich one acts by preference.: When &e place importance on something that &e cherish &e are $aluing that trait, ideal, or characteristic.!o%each, in his boo% titled 'he Nature of #uman Values, says there are t&o %inds of $alues that people ha$e >instrumental $alues> and >terminal $alues>. Instrumental $alues consist, primarily, of personal characteristic and character traits. 'erminal $alues are those things that &e can &or% to&ard or &e thin% are most important and that &e feel are most desirable. 'he follo&ing t&o tables illustrate e3amples of instrumental and terminal $alues.Instrumental Values Ambitious 7#ard0&or%ing, aspiring8 )roadminded 7Epen0minded8 6apable 76ompetent, effecti$e8 6heerful 7<ighthearted, 4oyful8 6lean 7Neat, tidy8 6ourageous 7*tanding up for your beliefs8 @orgi$ing 7Willing to pardon others8 #elpful 7Wor%ing for the &elfare of others8 #onest 7*incere, truthful8 Imaginati$e 7Daring, creati$e8 Independent 7*elf0reliant, self sufficient8 Intellectual 7Intelligent, reflecti$e8 <ogical 76onsistent, rational8 <o$ing 7Affectionate, tender8 Ebedient 7Dutiful, respectful8 Polite 76ourteous, &ell0mannered8 !esponsible 7Dependable, reliable8 *elf 0 controlled 7!estrained, self discipline8'erminal Values A &orld at Peace 7free of &ar and conflict8 @amily *ecurity 7ta%ing care of lo$ed ones8   @reedom 7independence, free choice8 "+uality 7brotherhood, e+ual opportunity for all8 *elf0respect 7self esteem8 #appiness 7contentedness8 Wisdom 7a mature understanding of life National security 7protection from attac%8 *al$ation 7sa$ed, eternal life8 'rue friendship 7close companionship8 A sense of accomplishment 7a lasting contribution8 Inner #armony 7freedom from inner conflict8 A comfortable life 7a prosperous life8 2ature lo$e 7se3ual and spiritual intimacy8 A &orld of beauty 7beauty of nature and the arts8 Pleasure 7an en4oyable leisurely life8 *ocial recognition 7respect, admiration8 An e3citing life 7a stimulating acti$e life8 Instrumental $alues in$ol$e &ays of being that help us arri$e at terminal $alues. 'hey are &ays of triggering our terminal $alues. 'erminal $alues are end states of feeling they are the emotional state that you  prefer e3periencing. 'erminal $alues ma%e our life fulfilling and &orth&hile. Instrumental $alues help get us there. A list of additional terminal $alues might include good health, po&er, passion, ad$enture, spontaneity, control, and many others.Where do our $alues come from? We tend to base our $alues on se$eral sources, namely, our culture, our  parental and familial influences, our teachers, friends, and other similar en$ironmental influences such as tele$ision, the internet and a host of other media outlets. 2ost of our $alues remain at the unconscious le$el. We don(t spend conscious time deciding if the things &e see and hear are $aluable to us. Interestingly, ad$ertisers fully understand this principle. If they promote an idea for a sufficient amount of time and &ith enough appeal, they can con$ince someone to belie$e in nearly any $alue. *mo%ing is a perfect e3ample of this. 'he people &ho produced tobacco reali5ed that their product is not $ery healthy choice and that people, if they really %ne&, &ould not be inclined to use their product. )ut &ith the right ad$ertising, &hich &as designed to con$ince us of the $alue of smo%ing, millions of people ha$e ta%en up a habit that most &ish they ne$er started in the first  place. If a person ne$er felt that there &as any $alue in lighting up this particular plant and stic%ing it in his mouth to inhale the fumes that come from the burning plant, he &ould ne$er choose to do it in the first place. 'he $alue he places on doing it al&ays determines &hether he &ill do it or not.We rarely +uestion most of the $alues that &e li$e by. @or e3ample, you might $alue the relationship you ha$e &ith your family &ithout really &ondering &hy. All of your life your parents ha$e espoused the
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