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Action algebra

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In algebraic logic, an action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure operation of the latter to the former, while adding the left and right residuation or implication operations of the former to the latter. Unlike dynamic logic and other modal logics of programs, for which programs and propositions form two distinct sorts, action algebra combines the two into a single sort. It can be thought of as a variant of intuitionistic logic with star and with a noncommutative conjunction whose identity need not be the top element. Unlike Kleene algebras, action algebras form a variety, which furthermore is finitely axiomatizable, the crucial axiom being a•(aa)* ≤ a. Unlike models of the equational theory of Kleene algebras (the regular expression equations), the star operation of action algebras is reflexive transitive closure in every model of the equations. Action algebras were introduced by Vaughan Pratt in the European Workshop JELIA'90.[1]

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  • Consciousness, a Quantum Physics Perspective
  • Lecture 17 - How to Design Hardware Products (Hosain Rahman)

Transcription

[MUSIC] Stanford University. >> So I'm Michael Heinrich. I'll be your host, essentially, for these nine weeks as we delve into the topic of consciousness. And particularly delighted to see such an interesting and varied audience, we've got ambitious Stanford students, Stanford professors. We even got some really interesting community members. We've got some Google engineers amongst us. We've got senior VPs and CEOs of Fortune 500 companies as well as startups. We've got meditators and non meditators. So a really interesting group of people, but before we get started I'd also like to sort of thank a few people that have made this possible. First our course advisors. One of which is Dr. Cesar Molina. He couldn't be with us today because he unfortunately had a travel booked before this course even was announced to him. But he did his residency at Stanford, he practices at the El Camino Hospital, and he's also the co-founder of The South Asian Heart Center, which really is all about prevention through lifestyle change, and sort of rest, diet, et cetera, is really part of their regimen. Another course advisor is John Bright, who is with us here today. Thank you John for all your support, and he is one of the leaders of the International Transcendental Meditation movement, TM for short. And of course Andrew Todhunter, who because of travel schedules is also not with us today, but he essentially is our core sponsor. He lectures in the Department of Biology and creative writing, and he is a really prolific writer, film maker, very, very creative, and really made this possible. And of course, the Department of Biology and the Head of Biology, Bob, for sort of making this possible and sponsoring it. So let me cover also actually, let me also thank the Stanford TM group for, sort of their support helping market this, and you guys are awesome. Thanks for always showing up for group meditation so thank you for, for your support as well. So before we get started, let me also cover a few logistical things. If you've signed up for the one unit course, you're allowed to miss one session. Since we only have nine sessions together, you know, ideally I'd like to see you as much as possible. I understand if sometimes you have some conflict, but if you wouldn't mind and essentially put down your name and also your student ID to make sure that I have a record of you being here, then I can sort of make sure that you've actually attended the class. And before we start too, do you have any questions about logistics, how the course will run, any questions at all? Good. Everybody. Oh, some question back there. >> Is there anything else required besides attending the classroom sessions? >> Ideally you do also the the reading, the required reading. So on every speaker we'll have their biography and usually some type of piece that they have either written or some piece written about them. And at the end, we also ask you to do a reflection piece, just a one page paper essentially about what you've learned in the class, or some piece of the reading, something like that. So, we really want to make it fun, so you can choose the medium. It could be a website if you wanted to, it could be just a paper. It's up to you really. Other questions? >> If we've already, already maxed credits, can we audit this course? >> Yes, you may audit this course. As I said, it's also open to the community because we invite sort of a really high quality audience, so that we can really push our speakers on this topic of consciousness and to really see, you know, how scientifically validated it is. You know, so it would be a lot of fun. So thanks for coming. Yes? >> [UNKNOWN] >> Yes, there's a syllabus, were you on the initial email? >> No. >> Okay, are you registered for the course, or. >> Not yet. >> Okay, so have you signed up for the Google form that I sent out? All right just leave your e-mail with me. [LAUGH] I will send you a syllabus. [LAUGH] All right, let me pass this through. Just pass that through,yeah. And if you're a student please sign in on this paper. And if you want you can also leave me your e-mail address in case you have not signed up for the Google form. Any other questions? Good, so let's dive into consciousness. So hacking consciousness, how did this come about? So I've always been fascinated by the topic of consciousness. As an undergraduate I studied cognitive science, and the predominant world view there is that this physical thing, the brain, this part of this, that's matter, gives rise to mental processes, and these mental processes are then considered consciousness. But I always found that answer somewhat unsatisfactory, because I asked myself, well, doesn't a flower have consciousness? Doesn't a tree have consciousness? I mean, there's some organizing power, right, to to a flower or to a tree. What is that intelligence that makes a flower bloom? And then I also I've always been really fascinated by mediation techniques. I've tried Qigon. I've tried mindfulness, Buddhist walking meditation, contemplation meditation, Zen meditation, and I always wanted to dive into this idea of consciousness. And what I noticed is that when I think a thought I can actually be conscious of me thinking a thought. So who is actually that observer behind that thought? And that's where sort of this whole interest came about in terms of hacking consciousness. And after undergraduate, I worked for a while and sort of forgot about this topic, until I found myself working for a man named Ray Dalio. I don't know if you guys are familiar with him, but he runs a company called Bridgewater Associates. It's one of the most successful macro hedge funds of all time. And he said that he's been practicing this meditation technique called transcendental meditation for 40 years. And he said that that is one of his keys to success. And I said, that's a big endorsement, so why not try it out myself. And through there again got exposed to this topic of consciousness and really found myself again experiencing as well as getting intellectual knowledge about it. So I thought, now that I'm at Stanford doing graduate studies, why don't I have other people participate in this journey and really see what consciousness is all about. Can you access it? Can you hack it? How do you hack it? And those are some of the questions that our speakers will address over these nine weeks. Which brings me to Dr. John Hagelin, who's our esteemed guest today. I could probably spend five minutes introducing his background, but I won't do that. [LAUGH] So I'll just mention some highlights. He is one of the most cited quantum physics, physicists of our time. He has written over 100 research publications about CERN He has done groundbreaking research at both CERN and SLAC, our very own SLAC. He's been part of many movies, including movies like What the Bleep do We Know. Has had many TV appearances. And this list goes on and on, so I'd rather just have you hear from John Hagelin himself. So please give a warm applause to John Hagelin. [APPLAUSE]. >> Thank you very much. [APPLAUSE] I'm not mic'd, can you hear me okay back there? >> Yes. >> It's great to be back here, sunny, campus looks wonderful. And what a great audience. I haven't even talked to you yet but just the description of who you are, should make this a lot of fun. Certainly for me, anyway. We're going to be introducing the subject of meditation basically from a very classic perspective from the Yogic tradition and from there ultimately the Buddhist tradition. And in the context of that talking partly from the standpoint of a fundamental physicist. What consciousness is or at least what we think it might be. And ultimately experiential access to it. Consciousness I'd have to say is really self hacking. But how do you hack it? So in the process I, I'm also an astrophysicist or a half astrophysicist. [LAUGH] I always like to spend a moment locating ourselves in the structure of the universe, and here of course are the, I almost said nine, eight planets. I grew up with nine, it's a sad story. And here they are in proper relative size to each other. They are certainly spaced farther apart than they look like in this. And our solar system is a pretty marvelous example of the solar system, we tend to be quite fond of it. It is our own and we do live here. That's what Saturn looks like, like Jupiter a gas colossus, a hydrogen and helium star. About a half a billion miles from the sun, about that distance from us as well. And it doesn't take much of a telescope in your backyard to see that incredible jewel out there like that. This is the ice world of Neptune. Now the last, the eighth of eight planets. That's what the noonday sun looks like from Neptune. A little like Iowa in the winter. You don't get much of a sun tan, there. And that's our solar system. Of course we're living in a galaxy, the Milky Way Galaxy with about 400 billion suns. Most of those we think, with some kind of a planetary system, not all. And in this sort of island universe, in this rather flat or pancake shape, and within our Milky Way galaxy which is that stripe you can probably see, the sky is good enough here that on a clear night you can see it. Incredible things happen, like stellar explosions called supernova, which are responsible, the final last gasp in the life of star that can result in this kind of cataclysmic explosion during which the heaviest elements get cooked, like gold and platinum. Anything heavier than iron, thrown out into space, contaminates space with heavier elements, because it really started as hydrogen and helium after the Big Bang. And from those heavier elements blown out into space, new stars form and planets form. Not just gas planets like Jupiter, but the Earth, made of dirt, silica, iron, et cetera, all come as remnants of supernova explosions. Our bodies comprised of things like, you know, oxygen, nitrogen, carbon were not cooked during the Big Bang. They were cooked in stars and then blown out into space from where our planets and our bodies were ultimately taken. If you take your back yard telescope and take it away from the plane of the Milky Way out in the direction of deep space, and if you have a good telescope you see this. You see more galaxies than stars. And each of these galaxies remember has several hundred billion suns, and you kind of get in the state of awe. A kind of a state of wonder. What enormous intelligence this is, what incredible creativity, what are the laws that govern it? And that's one of the quests of science and astronomy, all science actually to understand the fundamental laws governing the universe. So that's it for outer space. We're going to spend the rest of our time looking at inner space. The structure of reality at deeper and deeper levels. That means in the language of physics at least smaller and smaller spacetime scales. And one are the things we've learned, one of the most important things we've learned from the last hundred years or more of physics, is that the universe is structured in layers. Structured in layers of creation, from superficial to fundamental, from macroscopic to microscopic. And that inward exploration started at a very surface level, three hundred years of classical mechanics dedicated to the study of macroscopic matter. But turned about ninety years ago towards the exploration of the deeper world of the atom. Atoms and molecules, the world of quantum mechanics. And what we discovered is a little surprising and probably not what you learned in high school. Maybe what some of you have learned in college. And that is that the atom is not a billiard ball and not a solar system in miniature. In high school they maybe say that well you know, the sun is like the nucleus and the planets are like the electrons orbiting the nucleus. It's a quaint picture but it has absolutely nothing to do with reality. It serves no purpose and it doesn't work at all. So quantum mechanics is a whole new language. A whole new logic, a whole new mathematics that is subtler, more expansive, more profound, and appropriate to describe physics at this more fundamental level. Very soon physics went on to discover the atomic nucleus, to explore the nucleus and the fundamental particles that comprise the nucleus, quarks and leptons. And a whole new language was needed for that. Not a radical departure, but a substantive departure called quantum field theory which is the marriage of Einstein's relativity with quantum mechanics. Again though, a whole new formalism, a whole new mathematics required. And then ultimately the, certainly the hot topic in physics today as it has been since the super string revolution swept through here about 20 years ago is, maybe 25 years ago starting with super gravity theory, is unified field theory. Which in today's, you know, more common parlance is typically in the form of super string theory and M theory. These theories are very exciting for reasons that I really wish we could get into in great depth, I'll mention a few things. These theories fulfill Einstein's life long quest to discover the unified source of the diversified universe. The unified fountainhead of all the laws of nature that govern the functioning of the universe at every level. The mathematics of this is big, is extremely daunting, even though the fundamental reality is one of absolute simplicity. The mechanics of the expression of that fundamental unity into the diversity of the laws of nature is a rather complex, very non-linear process involving self interaction in a very non-trivial way. But speaking qualitatively instead of heavily mathematically, the fundamental nature of this field of unity at the basis of all diversity is absolute silence. Absolute abstraction, almost. You could say pure existence, pure being. But it's not a field of inertia. It's not a field of death, I would say, because the quantum principle guarantees that intrinsically this field is dynamic. In fact, the quantum principle, or uncertainty principle, as it's known, is the principle of increasing dynamism, at fundamental scale. The same reason nuclear power is more powerful than chemical energy, even though in some respects it's very similar. But it operates at a level that is a billion times smaller. And therefore, operates at a level of nature's dynamism that is a million times more powerful. Due to this intrinsic dynamism of this universal field. The field is actually, well you can say silent at its core but it's roiling and boiling on the surface of life, erupting in this effervescence of, of what looked like effervescent bubbles emerging from ginger ale. That's sometimes called zero point motion. And these days it's called dark energy. Because it's this type of activity that is driving the universe today in this ever accelerating expansion that has been relatively recently discovered. Now, for the astute audience, and most of you are certainly in that category, you probably have surmised that these bubbles aren't ginger ale actually. But they are, these little rubber bands, these little loops, they're infinitesimal loops called super strings. Little rubber bands, literally relativistic rubber hands, infinitesimal. And I know it's kind of depressing for those that are seeking the ultimate nature of reality to conclude that after all of these years of profound philosophy and scientific research we discovered that the core reality of the universe is a rubber band, but it's not really what we're saying. It's not the rubber band that is fundamental. It's this universal field of intelligence. This intrinsically dynamic self interacting field that percolates super strings, and these super strings are what we used to think of as particles, the fundamental particles in forces of nature, in the context of these theories, are just the different vibrational states, of these rubber bands. Now there's a certain type, you can sit down. And you can examine the physics of a rubber band. You can count the different vibrational states, the different modes of vibration that a string, a rubber band, can vibrate in and you will conclude that each of those vibrational modes of a rubber band has its own frequency, it's own natural tone. That type of mathematical physicist that undertakes this calculation of the enumeration of the vibrational frequencies of rubber bands are called nerds. [LAUGH] But they are very, very. But the miracle of super string theory, just honestly simplify things quite a bit for the sake of time. Is that these different vibrational tones, these different vibrational frequencies of these fundamental super strings, each frequency corresponding to a different energy. 'Cause Einstein equates, well, frequency of course, and energy and also Einstein equates energy with mass. These different rubber bands, due to their different vibrational modes,have different amount of energy, which means different amount of mass and when you look at those masses and the other properties of these vibrating strings, lo and behold, you discover. That this is a universe of just certain types of particles. One of those vibrational tones acts like gravity. A particle of gravity, a graviton. Another one acts like a particle of light. The spin one massless photon. Another one looks like a spin one half cork or an electron and so forth. And you get just. The categories of matter and energy that we know and love that are the building blocks so to speak of the universe. So, that's kind of an amazing prediction you start with a relatively simple principle I say deceptively principle deceptively simple to be fair, of a vibrating relativistic string. And from that you conclude that the universe coming out of such a field percolating these screens, looks like our's. Or at least, at least roughly like our's. It's a very different view of the universe. It's almost like this universal field, like a guitar string is a one dimensional field. Surface of a pond is a two dimensional field. This is a, a three or more dimensional field. But fields have their own, vibrational states, call them waves. And depending upon how the field is vibrating, depending upon the, the frequency of the wave, that wave, will behave like a graviton, a particle, a so called gravitino, a spin one force field, a spin one half fermion, or the newly discovered Higgs Boson. There's an interesting relationship which we can't even begin to touch on today, but these five actually correlate in a very precise way. To one of these ancient you know, these quaint ancient Prosaic theories of the universe is comprised fundamentally of five elements, and that's an interesting point perhaps for another day. Finally, I'm going to leave physics in a moment. I'm sure you'll be relieved. In addition to these super strings that percolate from this time translationally invariant it's this sort of ocean of immortality, ocean of pure being, are not just these superstrings and not just particles but entire universes. To the current ability of ours to calculate in the context of such theories there is a finite probability that whole baby universes will emerge from this bubbling cauldron of what is called space-time foam. And most of those are duds. They disappear almost immediately in a burst of energy, but given the right initial conditions, they grow, some of them, will grow exponentially. Expand enormously. And that is called inflation, the inflationary universe, big bang theory. If you look at this picture carefully you will see that there are several of these going on as we speak and some of those survive. And if you, you know, do the math even relatively simply you would conclude that there is. Depending upon a few assumptions, probably, an uncountable infinity of simultaneously coexisting universes continuously erupting from this enormous universal ocean of intelligence and that is called today the multiverse. A scary concept; I'm not in love with it. But is a concept that is sort of getting forced upon us as we look more deeply into the nature of this fundamental physical reality, and start to fathom it's intrinsic creativity and dynamism, and incredible properties. End of physics and I would like to shift gears. But, this will be I think a useful background. Any questions just on the physics giving the limited time we have I'd be happy to take one or two. This will be part of the test at the end of the course. (LAUGH). In this context, in this physical framework, what is consciousness and what is meditation? Now here, I'm going to draw upon two sources and a lot of what I say is not familiar or that familiar or yet familiar to, Western psychological science. Familiar, certainly, to some people within it but it's not really common parlance in Western psychological science. Consciousness is structured human consciousness, the one we can talk most clearly about, is structured in layers in parallel to the structure of the physical body, in parallel to the structure of the physical universe. Wow, what does that mean? Well, just subjectively, it means we have surface thoughts, concrete thinking, gotta do this, gotta do that, almost an audible level of coarse thinking, preverbal, in some cases even verbal. But quieter than that, in a more subtle, quiet, and expansive frame of mind,. Is the world of, of abstract concepts. The world, you could say, of the mathematician, the scholar. And this is a quiet and a literally more expansive style of thinking, with it a rather different character. And I'll talk about that. And even deeper, more abstract, more refined, more silent levels of thought. In these different levels of thought have a correspondence that can be rigorously unfolded to different levels of physical nature. And here's a simplified a truncated this argument but mathematics is probably the most, the most successful formalization of the structure of human mind. Structure of human thought structure of human logic. In mathematics comes in different layers of concreteness versus layers of great subtlety, greater power, greater comprehension, greater completeness, and if we just to make this idea, familiar and it probably is to some of you, we can start with the natural numbers, the counting numbers, one, two, three. These are the number we learn first. These are the numbers that have most concrete meaning and most concrete relevance in practical living. You go to the store. You buy three apples, not pie. Not the square root of negative one. But, you know, one, two, three apples. And that's, you know, it's own number frame work. Interestingly enough, if you add one more element to it called 0 you go from the natural numbers to the whole numbers. Not a big deal, you wouldn't think but it's kind of a big deal because the whole numbers are more complete it's a more powerful numerical system. There are things about the natural numbers that are true, but unprovable. Once you add these so called additive identities 0, you can prove things about the natural numbers that were always true, but unprovable before, because your numerical system didn't have sufficient power. All this power is at the expense of some concreteness. The number 0 is just a bit more abstract than the number three. A bit more abstract. You don't go to the store to buy zero apples. And it's not a number people that, use so much unless you're an accountant or something. They use so much in day to day living. If you had the negative numbers minus one, two, three, et cetera, you have what are called the integers. The integers are a more holistic, more powerful numerical framework. You now have a system of numbers that is closed under subtraction and in that respect its more powerful and you can prove a lot of things about the whole numbers that were true but unprovable you now have a more powerful frame making the expensive thing a bit more abstract to these you add the fractions point seven three two or two thirds or seven eighths. You have a more powerful system, a more comprehensive system, a system almost good enough to do physics. Not quite. For that you actually have to add not just the reals, not just the rationals, but the irrationals, in order to get the real numbers. And the irrationals fill in all the holes in between all the rationals. And they have the form of something like 0.732853 et cetera, without end. And they're difficult to even write down, obviously, and they're really, in a sense, somewhat difficult to even describe, but. With the system of real numbers you can at least do physics, you can do calculus, you can newtonian physics. But you add a deeper conceptual level, less concrete, much more powerful at the expense of being more abstract. Now, next step. Add numbers proportional to the square root of negative one. So called imaginary numbers it's a good word for them because I'm not going to even try to explain to you what they mean. They're really a giant step removed from practical day to day reality but enormously more powerful without these numbers the reals plus the imaginary equaling the complex numbers you cannot do quantum mechanics, you cannot understand the atom. Let alone the atomic nucleus. Let alone the unified field so different levels of mind different levels of conceptual wholeness relate to different more holistic but more abstract levels of nature these levels of nature are more abstract more powerful like that we have levels of mind. From this perspective meditation classically understood from the fundamental from historic yogic tradition, the [FOREIGN] and so on. And from there Buddhism at least early Buddhism. Meditation was understood as a technique to take our active thinking mind and our outwardly directed attention and turn the attention powerfully within to begin to experience and explore quieter levels of mind deeper deeper levels of the thinking process. And as the mind gets increasingly quiet and expansive, and quiet and expansive, very quickly or not so quickly, that's a question of technique, the awareness gets drawn completely beyond any localized concept or boundary of thought to experience a state of absolute. Abstraction. Pure subjectivity. Pure wakefulness, pure being, which is either a bunch of empty words or it has some meaning to you depending on perhaps whether you may have lived that experience at some time or other, in which case what I'm saying will make maybe a little more sense. This is the so called meditative state. This is sumati, not enlightenment, it's sumati. In the yoga sutra, the first couple of verses read. Yoga, experience of unity, inner union, union with what? Union with universal intelligence. Union of individual mind with universal intelligence. Yoga is the complete settling of the activity of the mind. Then, and automatically then, if you don't fall asleep, automatically the seer the experiencer is established in the self. And the self, or opman, in this position is not our time space bound selves, not our physical body, not our fluctuating thoughts, it's not our fluctuating moods, it's the field of our own inner subjectivity, inner weightfulness, inner consciousness. Which is at the basis of every experience but which itself is rarely experienced. So that's the meditative state, Samadhi, and it called, been known historically at least, described throughout the ages as a fourth state of human consciousceness. Distinct from waking, dreaming, or deep sleep. Those are the three cycling, relative states of consciousness. Samadhi has nothing to do with time, timeless state, non changing state. From a modern scientific perspective I think we can say since about 1970 and increasingly all the time this meditative state, and there are different states one can experience in meditation, I'm really talking more specifically amount Samadhi, classic mediation state. Slipping beyond thought all together to experience pure being. It is a fourth state of consciousness. It is metabolically distinct, and neurophysiologically distinct, and certainly subjectively, experientially distinct from waking, dreaming or sleeping. When the mind becomes absolutely still, the body simultaneously gains a state of deep relaxation or deep rest, significantly different from what is normally called "rest" or "relaxation" or even "sleep". Rest, I'm getting kind of practical here but trust me I won't stay practical for very long, rest of course is a very powerful antidote to stress. Relaxation is, you know, a powerful antidote to stress, and the different levels of rest you can achieve at night. Or through prayer, or through some meditation practice which perhaps can at least in principle really settle the individual, bring a sense of great comfort. To the degree that we're settling deeply, to that degree stress is dissolved and with it stress relateds disease are relieved and or prevented. So, for example, one very common technique to transcend, and that's really what tm technique is for, it's really for transcending, among other techniques, but, the transcending is the key, to, much of my discussion, not the technique for how to get there. But, the point is that state of transcending is a state of deep rest, and, for example, high blood pressure, which is most important risk factor for heart disease is ameliorated, more effectively than through hypertensive drugs, typically. And more effectively than what is normally called relaxation which is relaxing and generally good for you but maybe not as deep a state of physiological rest. The american heart association did the largest study ever done that was just this spring, excuse me late last spring, it's already 2014, and they looked at all kinds of alternative approaches to reducing high blood pressure and heart disease, and they, it was the best, the early research was done at Stanford. A really excellent meta analysis of many, many hundreds of published studies was, was done here, but the AHA just did another gigantic study, and they concluded, in a formal policy statement, that doctors should prescribe transcending. It mentions specifically TM for patients with blood pressure over 120 millimeters of mercury. Which I suspect is you know, a great, great many of you. Bottom line when it comes to, to, heart attack, I'm going to get off this, subject, he, of health soon because its kind of common sense. Most disease is caused by stress, or complicated by stress. If there is a really effective way to get rest deeper than sleep, deep rest at will. It seems plausible that you can unwind stress more effectively than mere sleep which helps enormously but doesn't always completely do the trick, and so this is a very interesting study. Nine year longitudinal study random control assignment study funded by the NIH which was just published in the last couple of years. Showing a two thirds drop in heart attack, stroke and death. In two random assigned groups, both at risk of heart disease, both groups took their medicines. Both groups supposedly stuck to their dietary and to their exercise regiments. The actual compliance for those was pretty poor, and one group though added transcending 20 minutes twice a day to their regimen and that group, that was the only difference, experienced a marked drop in heart disease. That was big news. There's more to health than heart disease and according to blue cross blue shield and their own statistics. Transcending is a very effective antidote to every category of disease on which they keep statistics, which is every category of disease. The only disease that wasn't reduced was childbirth, which is arguably not a disease. [LAUGH] The meditators were having just as many babies. And you put that all together, this is my last health related slide, stress and fatigue cause wear and tear on the system over the course of a lifetime and would you really stay up and cram for an exam maybe several nights in a row you start to feel the effects of aging and most of that will go away when you finally get some rest. But it does leave a mark. Deeper rest of transcending starts to really diffuse deeply seated stress, and the result of that is people who do this regularly take, you know, say, 20 minutes, twice a day, which is typically what an adult, recommended for an adult and your biological age, your, your, cardiovascular age and other ways that doctors can tell you how old you are, even if you lose your birth certificate. Five people are about 12 to 15 years younger than their chronological counterparts, just from meditating. There are a lot of other things you can do to improve health and longevity, but this is the easiest for sure. All right, I am an educator, and more interested in the brain, and brain development, and here I just have a couple of, just a couple of important things I think to say. Remember, from the perspective of today's, talk on meditation, and there will be others, because there are different things you can do. When you sit and close your eyes and practice the meditation program for different types of results and even for different reasons, and there are many ways to use the mind and many ways to develop the mind. Mathematics is one thing, well, anything you study is going to develop your mind and your mental abilities at least in specific respects, specific ways. So there are many things you can do to develop the mind. What I'm talking about today is the classic sense of meditation is defined for example of the yoga sutras is taking the awareness as efficiently as you can, beyond thought. This is not an automatic process. If you think about it, well how am I gonna quiet the mind completely, you know, forget it. It's just a matter of, of technique. If you, nobody ever told you how to go to sleep you may, you know, never fall asleep if you stare watching television all night long. But if they, look you know, turn off the TV, lie down like this and do this and do that and you'll probably fall asleep. It's like that with meditation. There's certain conditions you set up, certain simple techniques with the mind you can do and find the mind, being just lured, being just kind of sweetly lured to these these deeper, and deeper levels of mind which are so fulfilling. It's so intrinsically charming, so kind of fascinating that your mind just goes there. Just get a taste of the direction, you know when you walk through the airport and there's a Cinnabon somewhere and you kind of smell this thing, you just go for it. It's like that with the mind. Give it a taste of what it's like going to deeper levels of mind, and that's all the mind wants to do. So this is easy, it's just a question of like a diver. It's a question of how you leave the board, kind of the rest is automatic. Different levels of mind, surface, active, concentrating mind. Quiet states of self-reflection and beyond thought state of being. Each of these has a completely different style of brain functioning. The electrical activity of the brain, the activity that can be studied in many ways. EEG is a good approach to meditation because things change by the moment. Use something like PET or SPECT or MRI or FMRI, things, you know, you can capture things that are happening slowly but the EEG provides a window of things that are changing very fast. The whole signature of active thinking mind and concentration focus. is a lot of cognitive processing a lot of high frequency high amplitude gamma. In a meditation technique that focuses more on a state of quiet self sort of reflection self observation so called in the scientific literature typically open monitoring. Mindfulness is a very common popular form of this type of quiet. Self reflection and they're versions of that too. Notice a completely different state of mind in this active, concentrating, cognitive processing state, and then slipping beyond mental activity all together, beyond mind, to experience consciousness, has its own very interesting physiological state. So different levels of mind have their own levels of neuro-physiological activity, and the signature for this meditative state, for transcending, for samadhi. Is this a state of, of very high amplitude alpha coherence? Here's a picture of the whole brain front to back, and you see something that's quite unique to the meditative state, or to samadhi. You see the entire brain during experiences of the transcendent. The whole brain is functioning in concert. In a highly integrated, highly coherent almost synchronous fashion. If you process this mathematically and you look at what's going on in terms of the coherent functioning of the brain during relaxation versus samadhi. They're quite different. This is somebody in eyes closed relaxation, and these different dots are where the electrodes are placed on the scalp to look at the electrical firing of the neurons within the brain, and occasionally you see a bar connecting neighboring points measured on the scalp, and that means those two points are talking to each other. Those two parts of the brain are functioning in a correlated fashion. There's some kind of coordination, some kind of coherence in there, but not a whole lot, and in a meditative state, this is the same person three months later. They've learned to meditate. This is during the meditation experience. The whole brain is basically. Functioning in a completely integrated way, and that's remarkable. Remarkable for a brain guy because you don't see this in waking, dreaming, sleeping, hypnosis, anesthesia, or any drug induced state that I'm aware of. This, orderly brain functioning is not only philosophically or interesting from the Neuroscientist perspective, but it's actually very useful, because orderly brain functioning, let's call it is global EEG coherence. We all have some coherence. You would not be in this room today, if there were not some orderly cohered activity taking place in your brain. But that orderly brain functioning in the extent of orderly brain function it translates to, orderly thinking. Orderly thinking translates to orderly speech, coherent speech that translates to coherent, purposeful, effective action. Typically that means fulfilling action. But this orderly brain function the accorded research correlates with increasing IQ. Really? Increasing intelligence wait a minute, increasing academic performance. Learning ability short term and long term memory. Creativity according to test, alertness moral reasoning. Psychological stability emotional maturity. Everything good about the brain it turns out depends on its orderly functioning, and as an educator I've taught in many places the idea and the reality of having something you can do any student can do. Any adult in a nursing home can do, that was a very interesting Harvard study on the institutionalized elderly with TM and what happened to their memory, longevity, health. But the fact that, you know, there was something you could do that increases intelligence and creativity, and all measures of intelligence that are used within the field of education today are highly statistically significant improved by transcending. That's fairly remarkable. It would have maybe been considered impractical because everybody knows the brain you know, forges new connections and learn things very quickly, but sometime in the 30s, certainly by my age you have this precipitous and disastrous loss of raw intelligence. The pruning of the brain but now we know, and we've known now for probably 15 years that the brain is so plastic and so malleable and so capable of forging new connections and learning fundamentally new things really throughout life. But the problem with the brain, they also say is it's use it or lose it, and it's the use of the brain, in a very creative educational environment, that will develop certain competencies that you did not have before, and it's this specifically, it's the utilization of the entire brain in a highly integrated way. That correlates with intelligence, creativity more than anything else. So, that makes this idea, of experiencing Samadhi and then increasing the orderliness of brain function not just during cause who cares but after meditation is a very very significant finding. This is a stressed brain. A few walking around campus, everywhere, for that matter. And what stress does is really the opposite of what we really want in terms of brain functioning. Challenge is a good thing. Overwhelming challenge that causes stress and induces fight or flight response is not a good thing. That kind of stress shuts down the prefrontal cortex, the higher brain. Which is responsible for our higher human functions: judgement, planning, moral reasoning. And under chronic stress, unfortunately in certainly parts of the world are under chronic stress, mean in general life can be under chronic stress. Chronic stress shuts down the higher brain chronically. And to the extent the higher brain is shut down chronically, it's not developing. It fails to develop properly. And if it doesn't develop by the age of 25. It's not going to develop at all. And because of this pervasiveness of stress, this is something the Surgeon General has also been, been promoting recently, the, we're using, we have an under-utilization of the full resource of the human brain. The, he was, said actually we seem to be living in a society of arrested development. Meaning that you know, we're stuck in various stages of adolescence it's really at the age of 12 to the age of 25 that we really gain the full utilization really any utilization of the prefrontal cortex. The executive center of the brain. The CEO of the brain. And unfortunately, stress impedes the development of that. A few comments about acute stress that we get back to higher states of consciousness and hopefully to some questions. Acute stress and its manifestation as PTSD is the result of overwhelming trauma. Chronic or more typically overwhelming trauma. And what happens in that state is something called the amygdala, or fear center of the brain, gets overloaded and sometimes stuck on hyperdrive. It's like jamming the foot down on the accelerator of your car so hard that you break the linkage and you take your foot off the accelerator and it lies flat down. And your car is revving, it's not an ideal situation to find yourself in. Now normally rest, and relaxation allows these sorts of things to normalize. But unfortunately, for somebody often who's had that kind of trauma normal restive sleep does not get rid of that constant hyper vigilance and insomnia. And the tendency because the fear center, there's interest, we'll talk about it more. Fear center is on constant state of vigilance so that everybody, everything, every circumstance is perceived instinctively as a threat, and you're constantly because of that in a state of fight or flight. What's really exciting, it's some of the recent research that's been done, shows that the deeper rest of somhati, this meditative state, calms. It effectively deactivates the amygdala. And, in, a bit, about, 40% of the subjects in these studies that are funded by the N, by the, DoD. And by the Veteran's Administration are relieved of most of their symptoms after one medication. It's like, you have a computer that's gathered all kinds of crud, and you've, somebody finally tore, somebody says when's the last time you rebooted your computer? So I haven't rebooted it for months. I just put it to sleep at night. So that person will unplug your computer for you, whatever, and reboot it. And what happens is, you get a complete, you know, reboot. You're resyncing the whole system, refreshing the whole system. That's what somatics, like, that's what it does. It resets, resyncs, almost reboots. It's like you've come out of this completely new. It's like everything like a good nights sleep but more so, so incredibly fresh you know, you still owe $3000 to the bank but you feel completely fresh and you have a whole new perspective and a whole new capability of dealing with it. So that's amazing because of that I find this amazing I didn't know if I'd live long enough to see this. But militaries of the world, lots of them, and our own. Department of Defense, Veteran's Administration, and military academies like Norwich University are incorporating TM into military training as a, an antidote, a vaccine against the ravages of war stress. Many educators are very familiar with a phenomenon called ADHD. It's also a stress-related learning disorder. In the research, calming or restoring balance to brain functioning and integrated brain functioning, the research is really impressive. That the, the very, very deep rest, it's probably more than just the deep rest, that helped dissolve the angst and the stress that tends to fuel this condition. But it's the resynchronization of the brain and the reintegration. It's the higher brain, the prefrontal cortex that controls our attention. It's the controller of the attention, it's the executive of the brain, and if that part of the brain has been shut down, a person has very little control over their attention, these kids are bouncing off the wall. First question you might ask is, are they going to be able to meditate? Even close their eyes? Instinctively, the teachers would say no. But you know, an efficient technique for transcending which starts the mind just traversing in the direction of more expansion, more happiness, more satisfaction. These kids are just gone. You know, after like a minute, you know, they're just in there and after ten minutes, you know, somebody has to say. Ten minutes is up. For someone of that age, ten minutes of meditation is enough. They don't have the crust of stress that the rest of us do. So those results are amazing, and as we saw on the research on that, schools around here, and all over the country, and all over the world are incorporating transcending, a simple technique for transcending called TM into the curriculum and we're talking about a million students now at three hundred and fifty schools. I did not think this would happen three years ago. I would have said, you know, no way I'm kind of an optimist, but when it comes to schools, parents, religion. You have to really make a solid case for this, that it's not a religious practice, that it may have religious or spiritual implications, but that is not what the kids are taught. They are taught a technique, pure and simple, to take the mind effortlessly, to a place where they're going to be really able to relieve stress and come back in to class alert, focused, and revitalized. That's an amazing thing. Started really, not surprisingly, as a Midwesterner, I will say not surprisingly in San Francisco but it has now swept the world. In a different talk, this is our friend Jack Welch, e could talk about strengthening the executive functioning of the brain. I'll skip by that, I do want to end before we take discussion. Back where I started, and that is Samadhi and higher states of consciousness. We talked about this meditative state. A state of absolute silence. You could say almost absolute abstraction, absolute expansion. Nothing of a specific content to color or delimit the awareness. It is almost, it is, it's attributeless, in being so completely non-specific, there's nothing to give it any sense of time, or change, or relativity. The structure of this fourth state of consciousness. Which I should say maybe has been considered in recent history, not in early history, but in recent history, difficult to achieve. It is absolutely not patanjali's point of view in the yoga sutra's of patanjali. It's a four, four, basically four chapters. They're all dedicated to transcending and the third and fourth chapters are dedicated exclusively to Samadhi. It's not difficult, but I suppose you can always get in the way of doing so like you could get a way, in the way of going to sleep. If you really you know, squirm and struggle enough. You won't fall asleep. It's really like that in a sense with transcending. The structure of Samadhi is very simple. In the process of taking the awareness within, which means taking the outward directed attention systematically within to experience. And explore different levels of mind. And the deeper levels of mind actually, the awareness is more expanded. It's like when we focus on something which is really the opposite of meditation, the way I'm defining it. Focusing on something like this is is a sort of a localization of the awareness on to something very relative and very concrete. In meditation the awareness is basically retiring from that sharp focus, in the process of retiring from that sharp focus there's nothing so concretely stark to localize the attention. So the attention, there's less to localize, becoming more diffuse, more unlocalized, more unbounded. And then we the object of thought, a mantra is typically used for this purpose, specific type of mantra called a transcending mantra is used for this purpose. Then the awareness is completely not bound by anything. So, in that state you have experienced. You are immersed in the experience. Of our purely abstract maximally expanded silent self, at the expense of everything else, everything else, for that moment. And this is usually experiences of relatively fleeting few seconds and it does become more stable, more accessible with. A little bit of culturing and practice but for that moment you've given up everything so don't tell your mother write your mother saying I'm really getting interested and and really just you know experiencing Samadi and that's what I'm going to do. I'm going to meditate all day long and they will say will who's going to support you who's going to support me? This is a nonfunctional state. It is intrinsically satisfying, it's intrinsically blissful in the most sort of sublime and subtle and expansive way. But it's a nonfunctional state. It's called, considered to be a stepping stone to something more significant. Before I leave it though, let me just take a few quotes. In this case, from the Upanishad about this fourth state of consciousness. The fourth condition, fourth state of consciousness is Atman, the self in his own pure state. The awakened life of supreme consciousness. Also, from the Upanishads, It is neither this nor that, neither inner nor outer. Nor semi-consciousness, nor sleeping consciousness. It cannot be seen or touched. It is above all distinction, beyond thought and ineffable. In the union, with that, is the supreme proof of his reality, very important point the philosophers of science, and philosophers of consciousness, is peace and love. So, there has been debate. And probably is debated across this campus and across centuries as to whether consciousness really exists. You never see it, indirectly you do, 'cause it's the light of consciousness within that allows us to see everything else, but the consciousness itself is hidden, completely overshadowed by the content of experience. And because of the fact that you can't smell it, can't taste it, can't see it, and typically don't even experience it. Unless you happen to kind of, by accident or by practice, escape the confines of our constantly changing experience to instead, for a moment, be left free. For consciousness to experience itself. That's Samadhi. Then you can say, oh, oh, I was not experiencing any thing, not absorbed in any thought, any feeling, any idea, and yet, I was beyond time. I go on forever. That's the real proof of it. Otherwise you can argue it, but honestly the experience of it is, it's not really, it's beyond intellect. The very nature of intellect, we can talk around it, we can point to it. But the nature of intellect is to discriminate. To distinguish this from this. It is, it is dual by its nature, by drawing distinctions. Whereas consciousness is just pure, unity. Now, moving on to the last but important point. And that is, you know, why meditate? Well, today the medical science will give us many reasons why we should meditate. Educators who have done research will give you many reasons why. It's good for your academic achievement, good for your executive functioning. But traditionally that was not how meditation was sold. Mediation was for the experience of samadhi, know thyself, by direct experience of the self and more important than just glimpsing it live it and living it is called a classically, traditionally called enlightenment. So, how do we define enlightenment in this lesson today well in one sense it's this maximum orderliness and expansive state of brain function maximum EEG coherence stabilized. Which means not just during the meditative state- But when you're out in activity, dynamically engaged in activity, whether it's a sports competition, or a computer game, or an exam, that orderliness of brain functioning, the inner calm, inner silence, inner stability is stabilized. Come back to that, it's kind of important. Here's a quote. This is about enlightenment, it's sometimes called nirvikalpa samadhi, which means unbroken samadhi, continuous samadhi, or nitya samadhi which means eternal samadhi. This state of yoga samadhi, experience of unity, becomes a well founded state, an established state. Whether it's been respectfully and uninterruptedly cultured for a long time. Well, is that true? Yes, scientifically it appears to be true. I'm going to give you a, a quick, very quickly, a look at the brain during meditation of a relatively new meditator and somebody was been doing it twice a day for a number of years, eight years in the case of this particular subject one of the hundreds of subjects in this study. And if you go to a brain and you try to compare it with what's going on in this relatively new meditator and somebody whose been experiencing Samadhi regularly for sometime, be hardpressed to find it difference. Really, mathematically if you crunch the numbers it's not a whole lot of difference, it seems like transcending is transcending. Samadhi is Samadhi, whether you're experiencing it for the first time or the 5th time or the 500th time, the difference is outside of meditation, after meditation. And now what you find, is that this orderliness of brain function, in a state of absolute clarity and coherence, inner silence, it dissipates very quickly when your eyes open up, and you, and you realize you're late for class, and all that. But in the longer term meditator, this alpha coherence in across the entire brain, remains with you, inactivity. So now the whole brain is functioning coherently, but instead of doing nothing, it's engaged in a task but the brain resources are all utilized in virtually everything you do. Not just it turns out during dynamic activity but even during sleep. It's just interesting enough to point out. First some quotes from the Yoga Vasishta about enlightenment. He is awake but enjoys the calmness of deep sleep. Or he is awake in deep sleep. So when this inner light, you could say, Samadhi turns on, at that point, at some point it's inextinguishable. And you could be in the midst of a dynamic dream, but now witnessing and enjoying that dream from kind of a cosmic vantage point. Of unbounded inner immovability. Or under the knife during surgery. Under anesthesia, absolutely out cold, like a rock. But the inner awareness, inner light of consciousness, is not extinguished by that. Time passes instantly, because there's no gears turning. No cognitive processing in the brain that are ticking. Off time. So it's a very quick process of going in and going out. But during that whole time the continuum of the unboundedness or univ, universality of the self persists. And that's what the research shows in longer term meditators that during the high amplitude. Delta waves of deep sleep superimposed, is the high amplitude alpha coherence of Samadhi in one state that you could call enlightened sleep or witnessing sleep. Couple more points about enlightenment, because it's a very important subject and I may not get to see you again during this short course. Why is it called liberation, self realization? Self realization because by direct experience you realize the core nature of the self as basically beyond time. If that's the nature of the self, huge, expanded, universal, cosmic. To really know that, with any confidence, experience, experiencing until you get it. But, liberation need something a little bit more. So, Budist, kind of a budist term, but I think it's a bit accurate term to be used in this type of description. Let's look at the structure of enlightenment. In the state of enlightenment, you have the inner experience of absolute silence, the self as established within itself, as a silent witness to the dynamic change blowing on the surface of life. But in this case you haven't sacrificed everything, you're back in your waking life. Whether you've got an exam to take, whether you have a race to run, whether you have a book to read, all of that surface activity of thought, speech, action, is taking place not at the expense any more of the self. Established in the self, you are engaged in action. And the reason this is called liberation is because, if you erase this, you turn out the light. Which means you somehow forget the nature of the inner self. That all you're left with is pretty much life as we know it. The changing relative. And the changing relative is filled with. Change is filled with ups and downs, successes and failures. The Buddhists take a rather extreme point of view on this sometimes. They call that word, world, the world of suffering. Certainly because it includes suffering, no doubt about it, that is the experience of many locked in circumstances in which they are locked. But the Buddhist would go so far as to say even getting that, you know even, you know, spending buying some wonderful thing is ultimately sorrow because even as you're spending it you're realizing only 100 dollars remains. Or that wonderful thing is starting to rust already. And so forth. So the Buddhist would go so far as to say the whole world of activity is ultimately a world of, of suffering. I don't think most of us experience it that way, except from time to time. But the principle is at least true. As long as we are completely absorbed in our relative changing world. We are a bit like a football of circumstances that our emotions tend to be buffeted up and down by a good hair day or a very bad hair day. [LAUGH] Liberation means you can be in the midst of a bad hair day or good, for that matter, and it is really such a surface. Phenomenon, even something we'd consider to be pretty important in comparison to that, the origin of universes. You ought to think of it from a physicist perspective. An intelligence that is so expansive, in which individual awareness unites with that same fundamental intelligence that percolates strings. And universes that honestly from that natural perspective. It's not an intellectual mood we're making just from the experience of the continuum of silent contentment within. Nothing frankly is so important that it overshadows, overthrows your equanimity. Leaving you free, basically, to engage fully. The house is on fire. You get in there and you do it. But at the same time, you are completely, you're completely stable, and silent, and collected within. So here's a quote about that. In the state of, permanent Samadhi, he is in his own being. Pure, never changing, never move, never moving, unpollutable. And in peace, beyond desire. He watches the drama of the universe even though fully engaged in action, not a sense of like reclusiveness, that's a misunderstanding of the nature of enlightenment, fully engaged in action he does not act at all. And this is a Buddhist quote, quite recognizably so, established in the self one overcome with sorrow and suffering. Here's one last point I want to make. DO I have time for one last point? This idea of thought for intention having power. It's this idea we can't seem to get rid of. Does thought have power? When i was growing up it was a book called. The power of positive thinking. Anybody old enough to remember that booK? More recently it was the secret. But if you look back throughout, even scriptural history, this idea that intention has some kind of manifesting power has been win, with us for a long time. Of course we know to some degree, intention has manifesting power. It's our intention and motivation, that gets us out of bed, to go and achieve what have to achieve. So, of course mind, you know, has power, but this means something different. This, the idea is just, holding the intention, has some, strength of it's own. Some manifesting or precipitating strength of its own. I was the token scientist chosen to go on an Oprah show which is about this very popular book and I said yes I would come but there's something about the book I don't fully agree with and I feel obligated to explain it. Otherwise, you're going to have a lot of frustrated guests, a lot of frustrated viewers. And I said that, you know, The Secret, they're, in the book you'll see testimonials which I believe are valid. These are sincere testimonials of miraculous results. You put the sticker on your refrigerator that says pearl necklace and then twelve months later it came, that very same one I wanted. There are a lot of stories like that and there have been thankly throughout time, but you probably find there are more people that put the sticker on the refrigerator and dutifully remind themselves of that desire that they're supposed to keep nourishing. In a month, you know, a year later, they're very frustrated because they're no closer to the pearl necklace than they were before. And just, lo and behold, three days before the show, Oprah cancelled the show, saying they had too many guests that were disappointed by the book. I said, well that's too bad, because I, I could have explained why it works [LAUGH] and why it doesn't and perhaps how to make it work. We've already talked about. I don't have to explain anything more about it. There are different levels of mind. Different levels of mind have greater and greater conceptual power. And even greater physical power, physical energy because each of these levels of mind, corresponds to deeper and deeper levels of physical reality. Now the correspondence between these two can be very sharp. We've talked in this course, briefly, classical mechanics, quantum mechanics, quantum field theory, unified field theory. These terms are as familiar from Western psychological science but in the literature of yoga and the Vedic literature and Buddhist literature and so forth there are these four levels of mind, Vaikhari, which is almost articulated thought, surface though, abstract conceptual thought fine feeling level. Level of finest refined feeling which is very deep and very important to the quality of our life. This is where an artist, a fine artist might spend most of their time and then there's Samadhi, this correspondence between levels of mind and levels of physical reality, is very deep and I gave a little bit of a hint at it earlier today. But again, you know this meditative process is a process of getting more and more intimate, be familiar with deeper and deeper levels of thought. Where thoughts are more powerful. So, somebody might ask me, it's certainly been asked many times. Does prayer have power? And I can say oh, well I'll answer this as a physicist and as a meditator. In my experience it depends. I went to a church service in South Central LA. It was a very dramatic service of worship, and people were actually leaping out and shouting out the name of God. And it was kind of exciting, you know, on a surface level. It was really kind of an exciting experience. But there are deeper, I think more profound traditions of worship where you might go and experience you know, God's presence at a fine level of feeling and. Perhaps you have a more pervasive effect on the physiology. Maybe the environment. And then there's the idea, won't call it prayer, but, you know, from a religious perspective you might think of it as prayer. Taking the mind beyond thought to identify with universal intelligence on the level of pure being. That's the level from where thoughts emerged first as a fine impulse and they take up more concrete shape as they work their way through the machinery of thought but it's at that leve where a mustard seed this is a tiny impulse of thought that i suppose in principle could move mountains. So, the secret behind the secret I'd have to say is transcend. I'm going to stop there. A provocative statement was made by one very impressive person, is that life begins not at 40, thank goodness or I've already missed it, 50, not even 60. But life really begins at enlightenment and what that means is that enlightenment is when we're permanently aware of the field of pure life, pure awareness, pure vitality. And until that experience, we're living, in a sense, indirectly only. The experience of consciousness, the experience of life, at is, as it is reflected in our experience. But the field itself, which is absolute contentment, absolute expansion, is mist. SO in one literal sense at least, the field of life becomes accessible momentarily during meditation during semi. It becomes accessible permanently during sleep even during anesthesia, god forbid, that becomes necessary. Permanently establishes at the south comes regular emersion. Alternated with activity. Regular submersion in the south, alternated, and that's all it takes. And it takes a little time. But it's not a thing to be impatient about because the very process of meditation and tasting it, brings back it's, really from day one. So that's what I wanted to say today. Thank you. [NOISE] For more please visit us at stanford.edu.

Definition

An action algebra (A, ∨, 0, •, 1, ←, →, *) is an algebraic structure such that (A, ∨, •, 1, ←, →) forms a residuated semilattice in the sense of Ward and Dilworth,[2] while (A, ∨, 0, •, 1, *) forms a Kleene algebra in the sense of Dexter Kozen.[3] That is, it is any model of the joint theory of both classes of algebras. Now Kleene algebras are axiomatized with quasiequations, that is, implications between two or more equations, whence so are action algebras when axiomatized directly in this way. However, action algebras have the advantage that they also have an equivalent axiomatization that is purely equational. The language of action algebras extends in a natural way to that of action lattices, namely by the inclusion of a meet operation.[4]

In the following we write the inequality ab as an abbreviation for the equation ab = b. This allows us to axiomatize the theory using inequalities yet still have a purely equational axiomatization when the inequalities are expanded to equalities.

The equations axiomatizing action algebra are those for a residuated semilattice, together with the following equations for star.

1 ∨ a*•a* ∨ a   ≤   a*
a* ≤ (ab)*
(aa)*   ≤   aa

The first equation can be broken out into three equations, 1 ≤ a*, a*•a* ≤ a*, and aa*. Defining a to be reflexive when 1 ≤ a and transitive when aaa by abstraction from binary relations, the first two of those equations force a* to be reflexive and transitive while the third forces a* to be greater or equal to a. The next axiom asserts that star is monotone. The last axiom can be written equivalently as a•(aa)* ≤ a, a form which makes its role as induction more apparent. These two axioms in conjunction with the axioms for a residuated semilattice force a* to be the least reflexive transitive element of the semilattice of elements greater or equal to a. Taking that as the definition of reflexive transitive closure of a, we then have that for every element a of any action algebra, a* is the reflexive transitive closure of a.

Properties

The equational theory of the implication-free fragment of action algebras, those equations not containing → or ←, can be shown to coincide with the equational theory of Kleene algebras, also known as the regular expression equations. In that sense the above axioms constitute a finite axiomatization of regular expressions. Redko showed in 1967 that the regular expression equations had no finite axiomatization, for which John Horton Conway gave a shorter proof in 1971.[5] [6] Arto Salomaa gave an equation schema axiomatizing this theory which Dexter Kozen subsequently reformulated as a finite axiomatization using quasiequations, or implications between equations, the crucial quasiequations being those of induction: if xax then xa* ≤ x, and if axx then a*•xx. Kozen defined a Kleene algebra to be any model of this finite axiomatization.

Conway showed that the equational theory of regular expressions admit models in which a* was not the reflexive transitive closure of a, by giving a four-element model 0 ≤ 1 ≤ aa* in which aa = a. In Conway's model, a is reflexive and transitive, whence its reflexive transitive closure should be a. However the regular expressions do not enforce this, allowing a* to be strictly greater than a. Such anomalous behavior is not possible in an action algebra, which forces a* to be the least transitive reflexive element.

Examples

Any Heyting algebra (and hence any Boolean algebra) is made an action algebra by taking • to be ∧ and a* = 1. This is necessary and sufficient for star because the top element 1 of a Heyting algebra is its only reflexive element, and is transitive as well as greater or equal to every element of the algebra.

The set 2Σ* of all formal languages (sets of finite strings) over an alphabet Σ forms an action algebra with 0 as the empty set, 1 = {ε}, ∨ as union, • as concatenation, LM as the set of all strings x such that xML (and dually for ML), and L* as the set of all strings of strings in L (Kleene closure).

The set 2X² of all binary relations on a set X forms an action algebra with 0 as the empty relation, 1 as the identity relation or equality, ∨ as union, • as relation composition, RS as the relation consisting of all pairs (x,y) such that for all z in X, ySz implies xRz (and dually for SR), and R* as the reflexive transitive closure of R, defined as the union over all relations Rn for integers n ≥ 0.

The two preceding examples are power sets, which are Boolean algebras under the usual set theoretic operations of union, intersection, and complement. This justifies calling them Boolean action algebras. The relational example constitutes a relation algebra equipped with an operation of reflexive transitive closure. Note that every Boolean algebra is a Heyting algebra and therefore an action algebra by virtue of being an instance of the first example.

See also

References

  1. ^ Pratt, Vaughan (1990), "Action Logic and Pure Induction" (PDF), Logics in AI: European Workshop JELIA '90 (ed. J. van Eijck), LNCS 478, Springer-Verlag, pp. 97–120.
  2. ^ Ward, Morgan, and Robert P. Dilworth (1939) "Residuated lattices," Trans. Amer. Math. Soc. 45: 335–54. Reprinted in Bogart, K, Freese, R., and Kung, J., eds. (1990) The Dilworth Theorems: Selected Papers of R.P. Dilworth Basel: Birkhäuser.
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