Introduction

True history

The popular history of spaced repetition is full of myths and falsehoods. This text is to tell you the true story. The problem with spaced repetition is that it became too popular for its own effective replication. Like a fast mutating virus it keeps jumping from application to application, and tells its own story while accumulating errors on the way.

Who invented spaced repetition?

This is the story of how I solved the problem of forgetting. I figured out how to learn efficiently. Modesty is a waste of time, therefore I will add that I think I actually know how to significantly amplify human intelligence. In short: memory underlies knowledge which underlies intelligence. If we can control what we store in memory and what we forget, we can control our problem solving capacity. In a very similar way, we can also amplify artificial intelligence. Its a great relief to be able to type in those proud words after many years of a gag order imposed by commercial considerations.

Back in the early 1990s, I thought I knew how to turn education systems around the world upside down and make them work for all students. However, any major change requires a cultural paradigm shift. It is not enough for a poor student from a poor communist country to announce the potential for a change. I did that, in my Master's Thesis, but I found little interest in my ideas. Even my own family was dismissive. Luckily, I met a few smart friends at my university who declared they would use my ideas to set up a business. Like Microsoft changed the world of personal computing, we would change the way people learn. We owned a powerful learning tool: SuperMemo. However, for SuperMemo to conquer the world it had to ditch its roots for a while. To convince others, SuperMemo had to be a product of pure science. It could not have just been an idea conceived by a humble student.

To root SuperMemo in science, we made a major effort to publish our ideas in a peer-review journal, adopted a little known scientific term of " spaced repetition " and set our learning technology in a context of learning theory and the history of research in psychology. I am very skeptical of schools, certificates, and titles. However, I still went as far as to earn a PhD in economics of learning, to add respectability to my words.

Today, when spaced repetition is finally showing up in hundreds of respectable learning tools, applications, and services, we can finally stake the claim and plant the flag at the summit. Usership is going into hundreds of millions.

If you read SuperMemopedia here you may conclude that "Nobody should ever take credit for discovering spaced repetition". I beg to disagree. In this text I will claim the full credit for the discovery, and some solid credit for the dissemination of the idea. My contribution to the latter is waning thanks to the power of the idea itself and a growing circle of people involved in the concept (well beyond our company).

Perpetuating myths

It is Krzysztof Biedalak (CEO), who got least patience with fake news in reference to spaced repetition. I will then credit this particular text and the effort in mythbusting to his resolution to keep the history straight. SuperMemo for DOS was born 30 years ago (1987). Let's pay some tribute.

If you believe that Ebbinghaus invented spaced repetition in 1885, I apologize. When compiling the history of SuperMemo, we put the name of the venerable German psychologist at the top of the chronological list and the myth was born. Ebbinghaus never worked over spaced repetition.

Writing about history of spaced repetition is not easy. Each time we do it, we generate more myths through distortions and misunderstandings. Let's then make it clear and emphatic. There has been a great deal of memory research before SuperMemo. However, each time I give prolific credit, keep in mind the words of Biedalak:

If SuperMemo is a space shuttle, we need to acknowledge prior work done on bicycles. In the meantime, our competition is busy trying to replicate our shuttle, but the efforts are reminiscent of the Soviet Buran program. Buran has at least made one space flight. It was unmanned

This texts is to put the facts straight, and openly disclose the early steps of spaced repetition. This is a fun foray into the past that brings me a particular delight with the sense of "mission accomplished". Now that we can call our effort a global success, there is no need to make it more respectable than it really is. No need to make it more scientific, more historic, or more certified.

Spaced repetition is here and it here to stay. We did it!

Credits

The list of contributors to the idea of spaced repetition is too long to include in this short article. Some names do not show up because I simply run out of allocated time to describe their efforts. Dr Phil Pavlik got probably most fresh ideas in the field. An array of memory researchers investigate the impact of spacing on memory. Duolingo and Quizlet are leading competitors with a powerful impact on the good promotion of the idea. I failed to list many of my fantastic teachers who inspired my thinking. The whole host of hard-working and talented people at SuperMemo World would also deserve a mention. Users of SuperMemo constantly contribute incredible suggestions that drive further progress. The reward for most impactful explanation of spaced repetition should go to Gary Wolf of Wired, but there were many more. Perhaps some other day, I will have more time to write about all those great people in detail.

1985: Birth of SuperMemo

The drive for better learning

I spent 22 long years in the education system. Old truths about schooling match my case perfectly. I never liked school, but I always liked to learn. I never let school interfere with my learning. At entry to university, after 12 years in the public school system, I still loved learning. Schooling did not destroy that love for two main reasons: (1) the system was lenient for me, and, (2) I had full freedom to learn what I like at home. In Communist Poland, I never truly experienced the toxic whip of heavy schooling. The system was negligent and I loved the ensuing freedoms.

We all know that best learning comes from passion. It is powered by the learn drive. My learn drive was strong and it was mixed with a bit of frustration. The more I learned, there more I could see the power of forgetting. I could not remedy forgetting by more learning. My memory was not bad in comparison with other students, but it was clearly a leaky vessel.

In 1982, I paid more attention to what most students discover sooner or later: testing effect. I started formulating my knowledge for active recall. I would write questions on the left side of a page and answers in a separate column to the right:

This way, I could cover the answers with a sheet of paper, and employ active recall to get a better memory effect from review. This was a slow process but the efficiency of learning increased dramatically. My notebooks from the time are described as "fast assimilation material", which referred to the way my knowledge was written down.

In the years 1982-1983, I kept expanding my "fast assimilation" knowledge in the areas of biochemistry and English. I would review my pages of information from time to time to reduce forgetting. My retention improved but it was only a matter of time when I would hit the wall again. The more pages I had, the less frequent the review, the more obvious the problem of leaking memory. Here is an example of a repetition history from that time:

Between June 1982 and December 1984, my English-Polish word-pairs notebook included 79 pages looking like this:

Figure: A typical page from my English-Polish words notebook started in June 1982. Word pairs would be listed on the left. Review history would be recorded on the right. Recall errors would be marked as dots in the middle

Those 79 pages would encompass a mere 2794 words. This is just a fraction of what I needed, and already quite a headache to review. Interestingly, I started learning English in an active way, i.e. using Polish-English word pairs only in 1984, i.e. with two years delay. I was simply late to discover that passive knowledge of vocabulary is ok in reading, but it is not enough to speak a language. This kind of ignorance after 6 years of schooling is a norm. Schools do a lot of drilling, but shed very little light on what makes efficient learning.

In late 1984, I decided to improve the review process and carry out an experiment that has changed my life. In the end, three decades later, I am super-proud to notice that it actually affected millions. It has opened the floodgates. We have an era of faster and better learning.

This is how this initial period was described in my Master's Thesis in 1990:

The birthday of spaced repetition: July 31, 1985

Intuitions

In 1984, my reasoning about memory was based on two simple intuitions that probably all students have:

• if we review something twice, we remember it better. That's pretty obvious, isn't it? If we review it 3 times, we probably remember it even better
• if we remember a set of notes, they will gradually disappear from memory, i.e. not all at once. This is easy to observe in life. Memories have different lifetimes

These two intuitions should make everyone wonder: how fast and how many notes we lose and when we should review next?

To this day, I am amazed that very few people ever bothered to measure that " optimum interval". When I measured it myself, I was sure I would find more accurate results in books on psychology. I did not.

Experiment

The following simple experiment led to the birth of spaced repetition. It was conducted in 1985 and first described in my Master's Thesis in 1990. It was used to establish optimum intervals for the first 5 repetitions of pages of knowledge. Each page contained around 40 word-pairs and the optimum interval was to approximate the moment in time when roughly 5-10% of that knowledge was forgotten. Naturally, the intervals would be highly suited for that particular type of learning material and for a specific person, in this case, me. In addition, to speed things up, the measurement samples were small. Note that this was not a research project. It was not intended for publication. The goal was to just speed up my own learning. I was convinced someone else must have measured the intervals much better, but 13 years before the birth of Google, I thought measuring the intervals would be faster than digging into libraries to find better data. The experiment ended on Aug 24, 1985, which I originally named the birthday of spaced repetition. However, while writing this text in 2018, I found the original learning materials, and it seems my eagerness to learn made me formulate an outline of an algorithm and start learning human biology on Jul 31, 1985.

For that reason, I can say that the most accurate birthday of SuperMemo and computational spaced repetition was Jul 31, 1985.

By July 31, before the end of the experiment, the results seemed predictable enough. In later years, the findings of this particular experiment appeared pretty universal and could be extended to more areas of knowledge and to the whole healthy adult population. Even in 2018, the default settings of Algorithm SM-17 do not depart far from those rudimentary findings.

Here is the original description of the experiment from my Master's Thesis with minor corrections to grammar and style. Emphasis in the text was added in 2018 to highlight important parts. If it seems boring and unreadable, compare Ebbinghaus 1885. This is the same style of writing in the area of memory. Only goals differed. Ebbinghaus tried to understand memory. 100 years later, I just wanted to learn faster:

On July 31, 1985, I could already sense the outcome of the experiment. I started using SuperMemo on paper to learn human biology. That would be the best date to call for the birthday of SuperMemo.

The events of July 31, 1985

On July 31, 1985, SuperMemo was born. I had most of my data from my spaced repetition experiment available. As an eager practitioner, I did not wait for the experiment to end. I wanted to start learning as soon as possible. Having built a great deal of notes in human biology, I started converting those notes into Special Memorization Test format (SMT was the original name for SuperMemo, and spaced repetition).

My calculations told me that, at 20 min/day, I would need 537 days to process my notes and finish the job by January 1987. I also computed that each page of the test would likely cost me 2 hours of life. Despite all the promise and speed of SuperMemo, this realization was pretty painful. The speed of learning in college is way too fast for the capacity of human memory. Now that I could learn much faster and better, I also realized I wouldn't cover even a fraction of what I thought was possible. Schools make no sense with their volume and speed. On the same day, I found out that the Polish communist government lifted import tariffs on microcomputers. This should make it possible, at some point, to buy a computer in Poland. This opened a way to SuperMemo for DOS 2.5 years later.

Also on July 31, I noted that if vacation could last forever, I would achieve far more in learning and even more in life. School is such a waste of time. However, the threat of conscription kept me in line. I would enter a path that would make me enroll in university for another 5 years. However, most of that time was devoted to SuperMemo and I have few regrets.

My spaced repetition experiment ended on Aug 24, 1985. I also started learning English vocabulary. By that day, I managed to have most of my biochemistry material written down in pages for SuperMemo review.

Note: My Master's Thesis mistakenly refers to Oct 1, 1985 as the day when I started learning human biology (not July 31 as seen in the picture above). Oct 1, 1985 was actually the first day of my computer science university and was otherwise unremarkable. With the start of the university, my time for learning and energy for learning were cut dramatically. Paradoxically, the start of school always seems to augur the end of good learning.

First spaced repetition algorithm: Algorithm SM-0, Aug 25, 1985

As a result of my spaced repetition experiment, I was able to formulate the first spaced repetition algorithm that required no computer. All learning had to be done on paper. I did not have a computer back in 1985. I was to get my first microcomputer, ZX Spectrum, only in 1986. SuperMemo had to wait for the first computer with a floppy disk drive (Amstrad PC 1512 in the year 1987).

I often get asked this simple question: "How can you formulate SuperMemo after an experiment that lasted 6 months? How can you predict what would happen in 20 years?"

The first experiments in reference to the length of optimum interval resulted in conclusions that made it possible to predict the most likely length of successive inter-repetition intervals without actually measuring retention beyond weeks! In short, it could be illustrated with the following reasoning. If the first months of research yielded the following optimum intervals: 1, 2, 4, 8, 16 and 32 days, you could hope with confidence that the successive intervals would increase by a factor of two.

To this day I hear some people use or even prefer the paper version of SuperMemo. Here is a description from 1992.

Note that the intuition that intervals should increase twice is as old as the theory of learning. In 1932, C. A. Mace hinted on the efficient learning methods in his book " The psychology of study". He mentioned " active rehearsal" and " repetitive revisions" that should be spaced in gradually increasing intervals, roughly " intervals of one day, two days, four days, eight days, and so on". This proposition was later taken on by other authors. Those included Paul Pimsleur and Tony Buzan who both proposed their own intuitions that involved very short intervals (in minutes) or "final repetition" (after a few months). All those ideas did not permeate well into the practice of study beyond the learning elites. Only a computer application made it possible to start learning effectively without studying the methodology.

That intuitive interval multiplication factor of 2 has also shown up in the context of studying the possibility of evolutionary optimization of memory in response to statistic properties of the environment: " Memory is optimized to meet probabilistic properties of the environment "

Despite all its simplicity, in my Master's Thesis, I did not hesitate to call my new method "revolutionary":

Birth of SuperMemo from a decade's perspective

A decade after the birth of SuperMemo, it became pretty well-known in Poland. Here is the same story as retold by J. Kowalski, Enter in 1994:

It was 1982, when a 20-year-old student of molecular biology at Adam Mickiewicz University of Poznan, Piotr Wozniak, became quite frustrated with his inability to retain newly learned knowledge in his brain. This referred to the vast material of biochemistry, physiology, chemistry, and English, which one should master wishing to embark on a successful career in molecular biology. One of the major incentives to tackle the problem of forgetting in a more systematic way was a simple calculation made by Wozniak which showed him that by continuing his work on mastering English using his standard methods, he would need 120 years to acquire all the important vocabulary. This not only prompted Wozniak to work on methods of learning, but also, turned him into a determined advocate of the idea of one language for all people (bearing in mind the time and money spent by the mankind on translation and learning languages). Initially, Wozniak kept increasing piles of notes with facts and figures he would like to remember. It did not take long to discover that forgetting requires frequent repetitions and a systematic approach is needed to manage all the newly collected and memorized knowledge. Using an obvious intuition, Wozniak attempted to measure the retention of knowledge after different inter-repetition intervals, and in 1985 formulated the first outline of SuperMemo, which did not yet require a computer. By 1987, Wozniak, then a sophomore of computer science, was quite amazed with the effectiveness of his method and decided to implement it as a simple computer program. The effectiveness of the program appeared to go far beyond what he had expected. This triggered an exciting scientific exchange between Wozniak and his colleagues at Poznan University of Technology and Adam Mickiewicz University. A dozen of students at his department took on the role of guinea pigs and memorized thousands of items providing a constant flow of data and critical feedback. Dr Gorzelańczyk from Medical Academy was helpful in formulating the molecular model of memory formation and modeling the phenomena occurring in the synapse. Dr Makałowski from the Department of Biopolymer Biochemistry contributed to the analysis of evolutionary aspects of optimization of memory (NB: he was also the one who suggested registering SuperMemo for Software for Europe). Janusz Murakowski, MSc in physics, currently enrolled in a doctoral program at the University of Delaware, helped Wozniak solve mathematical problems related to the model of intermittent learning and simulation of ionic currents during the transmission of action potential in nerve cells. A dozen of forthcoming academic teachers, with Prof. Zbigniew Kierzkowski in forefront, helped Wozniak tailor his program of study to one goal: combining all aspects of SuperMemo in one cohesive theory that would encompass molecular, evolutionary, behavioral, psychological, and even societal aspects of SuperMemo. Wozniak who claims to have discovered at least several important and never-published properties of memory, intended to solidify his theories by getting a PhD in neuroscience in the US. Many hours of discussions with Krzysztof Biedalak, MSc in computer science, made them both choose another way: try to fulfill the vision of getting with SuperMemo to students around the world.

1986: First steps of SuperMemo

SuperMemo on paper

On Feb 22, 1984, I computed that at my learning rate and my investment in learning, it would take me 26 years to master English (in SuperMemo, Advanced English standard is 4 years at 40 min/day). With the arrival of SuperMemo on paper that statistic improved dramatically.

In summer 1985, using SuperMemo on paper, I started learning with great enthusiasm. For the first time ever, I knew that all investment in learning would pay. Nothing could slip through the cracks. This early enthusiasm makes me wonder why I did not share my good news with others.

SuperMemo wasn't a "secret weapon" that many users employ to impress others. I just thought that science must have answered all questions related to efficient learning. My impression was that I only patched my own poor access to western literature with a bit of own investigation. My naivete of the time was astronomical. My English wasn't good enough to understand news from the west. America was for me a land of super-humans who do super-science, land on the moon, do all major discoveries and will soon cure cancer and become immortal. At the same time, it was a land of Reagan who could blast Poland off the surface of the Earth with Pershing or cruise missiles. That gave me a couple of nightmares. Perhaps the only major source of stress in the early 1980s. I often ponder amazing inconsistencies in the brains of toddlers or kids. To me, the naivete of my early twenties tells me I must have been a late bloomer with very uneven development. Ignorance of English translated to the ignorance of the world. I was a young adult with areas of strength and areas of incredible ignorance. In that context, spaced repetition looks like a child of a need combined with ignorance, self-confidence, and passion.

University

In October 1985, I started my years at a computer science university. I lost my passion for the university in the first week of learning. Instead of programming, we were subject to excruciatingly boring lectures of introductory topics in math, physics, electronics, etc. With a busy schedule, I might have easily become a SuperMemo dropout. Luckily, my love for biochemistry and my need for English would not let me slow down. I continued my repetitions, adding new pages from time to time. Most of all, I had a new dream: to have my own computer and do some programming on my own. One of the first things I wanted to implement was SuperMemo. I would keep my pages on the computer and have them scheduled automatically.

I casually mentioned my super-learning method to my high school friend Andrzej "Mike" Kubiak only in summer 1987 (Aug 29). We played football and music together. I finally showed him how to use SuperMemo on Nov 14, 1987. It took 836 days (2 years 3 months and 2 weeks) for me to recruit the first user of SuperMemo. Mike was later my guinea pig in trying out SuperMemo in procedural learning. He kept practicing computer-generated rhythms using a SuperMemo-like schedule. For Mike, SuperMemo was a love at first sight. His vocabulary rocketed. He remained faithful for many years up to a point when the quality of his English outstripped the need for further learning. He is a yogi and his trip to India and regular use of English have consolidated the necessary knowledge for life.

ZX Spectrum

In 1986 and 1987, I kept thinking about SuperMemo on a computer more and more often. Strangely, initially, I did not think much about the problem of separating pages into individual flashcards. This illustrates how close-minded we can be when falling into a routine of doing the same things daily. To get to the status of 2018, SuperMemo had to undergo dozens of breakthroughs and similarly obvious microsteps. It is all so simple and obvious in hindsight. However, there are hidden limits of human thinking that prevented incremental reading from emerging a decade earlier. Only a fraction of those limits is in technology.

In my first year of university I had very little time and energy to spare, and most of that time I invested in getting my first computer: ZX Spectrum (Jan 1986). I borrowed one from a friend for a day in Fall 1985 and was totally floored. I started programming "on paper" long before I got the toy. My first program was "planning the day" precursor of Plan. The program was ready to type in into the computer when I turned on my ZX Spectrum for the first time on Jan 4, 1986. As of that day, I spent most of my days on programming, ignoring school and writing my programs on paper even during classes.

The Army

Early 1986 was marred by the threat of conscription. I thought 5 more years of university meant 5 more years of freedom. However, The Army had different ideas. For them, second major did not count, and I had to bend over backwards to avoid the service. My anger was tripled by the fact that I would never ever contemplate 12 months of separation from my best new friend: ZX Spectrum. I told the man in uniform that they really do not want to have an angry man with a gun in their ranks. Luckily, in the mess of the communist bureaucracy, I managed to slip the net and continue my education. To this day, I am particularly sensitive to issues of freedom. Conscription isn't much different from slavery. It was not a conscription in the name of combating fascism. It was a conscription for mindless drilling, goosestep, early alarms, hot meals in a hurry and stress. If this was to serve the readiness of Communist Bloc, this would be a readiness of Good Soldier Švejk Army. Today, millions of kids are sent to school in a similar conscription-like effort verging on slavery. Please read my " I would never send my kids to school" for my take on the coercive trample of the human rights of children. I am sure that some of my sentiments have been shaped by the sense of enslavement from 1986.

On the day when the radioactive cloud from Chernobyl passed over Poznan, Poland, I was busy walking point to point across the vast city visiting military and civilian offices in my effort to avoid the army. I succeeded and summer 1986 was one of the sunniest ever. I spent my days on programming, jogging, learning with SuperMemo (on paper), swimming, football and more programming.

Summer 1986

My appetite for new software was insatiable. I wrote a program for musical composition, for predicting the outcomes of the World Cup, for tic-tac-toe in 3D, for writing school tests, and many more. I got a few jobs from the Department of Biochemistry (Adam Mickiewicz University). My hero, Prof. Augustyniak, needed software for simulating the melting of DNA, and for fast search of tRNA genes (years later that led to a publication). He also commissioned a program for regression analysis that later inspired progress in SuperMemo (esp. Algorithms SM-6 and SM-8).

While programming, I had SuperMemo at the back of my mind all the time, however, all my software was characterized by the absence of any database. The programs had to be read from a cassette tape which was a major drag (it did not bother me back in 1986). It was simpler to keep my SuperMemo knowledge on paper. I started dreaming of a bigger computer. However, in Communist Poland, the cost was out of reach. Once I computed that an IBM PC would cost as much as my mom's lifetime wages in the communist system. As late as in 1989, I could not afford a visit in a toilet in Holland because it was so astronomically expensive when compared with wages in Poland.

PC 1512

My whole family pulled in resources. My cousin, Dr Garbatowski, arranged a special foreign currency account for Deutsch Mark transfers. By a miracle, I was able to afford DM 1000 Amstrad PC 1512 from Germany. The computer was not smuggled as it was once reported in the press. My failed smuggling effort came two years earlier in reference to ZX Spectrum. My friends from Zaire were to buy it for me in West Berlin. In the end, I bought second-hand ZX Spectrum in Poland, at a good price, from someone who thought he was selling "just a keyboard".

My German Amstrad-Schneider PC 1512 was ordered from a Polish company Olech. Olech was to deliver it in June 1987. They did it in September. This cost me the whole summer of stress. Some time later, Krzysztof Biedalak ordered a PC from a Dutch company Colgar and never got a PC or money back. If this happened to me, I would have lost my trust in humanity. This would have killed SuperMemo. This might have killed my passion for computers. Biedalak, on the other hand, stoically got back to hard work and earned his money back and more. That would be one of the key personality differences between me and Biedalak. Stress resilience should be one of the components of development. I developed my stress resilience late with self-discipline training (e.g. winter swimming or marathons). Having lost his money, Biedalak did not complain. He got it back in no time. Soon I was envious of his new shiny PC. His hard work and determination in achieving goals was always a key to the company's survival. It was his own privately earned money that helped SuperMemo World survive the first months. He did not get a gift from his parents. He could always do things on his own.

Simulating the learning process

On Feb 22, 1986, using my ZX Spectrum, I wrote a program to simulate long-term learning process with SuperMemo. I was worried that with the build-up of material, the learning process would slow down significantly. However, my preliminary results were pretty counterintuitive: the progress is almost linear. There isn't much slow down in learning beyond the very initial period.

On Feb 25, 1986, I extended the simulation program by new functions that would answer " burning questions about memory". The program would run on Spectrum over 5 days until I could get full results for 80 years of learning. It confirmed my original findings.

On Mar 23, 1986, I managed to write the same simulation program in Pascal which was a compiled language. This time, I could run 80 years simulation in just 70 minutes. I got the same results. Today, SuperMemo still makes it possible to run similar simulations. The same procedure takes just a second or two.

Some of the results of that simulation are valid today. Below I present some of the original findings. Some might have been amended in 1990 or 1994.

Learning curve is almost linear

The learning curve obtained by using the model, except for the very initial period, is almost linear.

New items take 5% of the time

In a long-term process, for the forgetting index equal to 10%, and for a fixed daily working time, the average time spent on memorizing new items is only 5% of the total time spent on repetitions. This value is almost independent of the size of the learning material.

Speed of learning

According to the simulation, the number of items memorized in consecutive years when working one minute per day can be approximated with the following equation:

NewItems=aar*(3*e ^-0.3*year+1)

where:

• NewItems - items memorized in consecutive years when working one minute per day,
• year - ordinal number of the year,
• aar - asymptotic acquisition rate, i.e. the minimum learning rate reached after many years of repetitions (usually about 200 items/year/min)

In a long-term process, for the forgetting index equal to 10%, the average rate of learning for generic material can be approximated to 200-300 items/year/min, i.e. one minute of learning per day results in the acquisition of 200-300 items per year. Users of SuperMemo usually report the average rate of learning from 50-2000 items/year/min.

Workload

For a generic material and the forgetting index of about 10%, the function of time required daily for repetitions per item can roughly be approximated using the formula:

time=1/500*year ^-1.5+1/30000

where:

• time - average daily time spent for repetitions per item in a given year (in minutes),
• year - year of the process.

As the time necessary for repetitions of a single item is almost independent of the total size of the learned material, the above formula may be used to approximate the workload for learning material of any size. For example, the total workload for a 3000-element collection in the first year will be 3000/500*1+3000/30000=6.1 (min/day).

Optimum forgetting index

The greatest overall knowledge acquisition rate is obtained for the forgetting index of about 20-30%. This results from the trade-off between reducing the repetition workload and increasing the relearning workload as the forgetting index progresses upward. In other words, high values of the forgetting index result in longer intervals, but the gain is offset by an additional workload coming from a greater number of forgotten items that have to be relearned.

For the forgetting index greater than 20%, the positive effect of long intervals on memory resulting from the spacing effect is offset by the increasing number of forgotten items.

When the forgetting index drops below 5%, the repetition workload increases rapidly (see the figure above). The recommended value of the forgetting index used in the practice of learning is 6-14%.

In later years, the value of the optimum forgetting index was found to differ depending on the tools used (e.g. Algorithm SM-17).

Memory capacity

The maximum lifetime capacity of the human brain to acquire new knowledge by means of learning procedures based on the discussed model can be estimated as no more than several million items. As nobody is likely to spend all his life on learning, I doubt I will ever see anyone with a million items in his memory.

1987: SuperMemo 1.0 for DOS

SuperMemo 1.0 for DOS: day by day (1987)

SuperMemo history file says " Wozniak wrote his first SuperMemo in 16 evenings". The reality was slightly more complex and I thought I would describe it in more details using the notes of the day.

I cannot figure out what I meant writing on Jul 3, 1987 that " I have an idea of a revolutionary program arranging my work and scientific experiments SMTests" (SMTests stands for SuperMemo on paper). A transition from paper to a computer seems like an obvious step. There must have been some mental obstacle on the way that required "thinking out of the box". Unfortunately, I did not write down details. Today it only matters in that it illustrates how excruciatingly slow a seemingly obvious idea may creep into the mind.

On Sep 8, 1987, my first PC arrived from Germany (Amstrad PC 1512). My enthusiasm was unmatched! I could not sleep. I worked all night. The first program I planned to write was to be used for mathematical approximations. SuperMemo was second in the pipeline.

Oct 16, 1987, Fri, in 12 hours I wrote my first SuperMemo in GW-Basic (719 minutes of non-stop programming). It was slow like a snail and buggy. I did not like it much. I did not start learning. Could this be the end of SuperMemo? Wrong choice of a programming language? Busy days at school kept me occupied with a million unimportant things. Typical school effect: learn nothing about everything. No time for creativity and your own learning. Luckily, all the time I used SuperMemo on paper. The idea of SuperMemo could not have died. It had to be automated sooner or later.

On Nov 14, 1987, Sat, SuperMemo on paper got its first user: Mike Kubiak. He was very enthusiastic. The fire kept burning. On Nov 18, I learned about Turbo Pascal. It did not work on my computer. In those days, if you had a wrong graphics card, you might struggle. Instead of Hercules, I had a text-mode monochrome (black-and-white) CGA. I managed to solve the problem by editing programs in the RPED text editor rather than in the Turbo Pascal environment. Later I got the right version for my display card. Incidentally, old SuperMemos show in colors. I was programming it in shades of gray and never knew how it really looked in the color mode.

Nov 21, 1987 was an important day. It was a Saturday. Days free from school are days of creativity. I hoped to get up at 9 am but I overslept by 72 minutes. This is bad for the plan, but this is usually good for the brain and performance. I started the day from SuperMemo on paper (reviewing English, human biology, computer science, etc.). Later in the day, I read my Amstrad PC manual, learned about Pascal and Prolog, spent some time thinking how human cortex might work, did some exercise, and in the late evening, in a slightly tired state of mind, in afterthought, decided to write SuperMemo for DOS. This would be my second attempt. However, this time I chose Turbo Pascal 3.0 and never regretted. To this day, as a direct consequence, SuperMemo 17 code is written in Pascal (Delphi).

For the record, the name SuperMemo was proposed much later. In those days, I called my program: SMTOP for Super-Memorization Test Optimization Program. In 1988, Tomasz Kuehn insisted we call it CALOM for Computer-Aided Learning Optimization Method.

Nov 22, 1987 was a mirror copy of Nov 21. I concluded that I know how cortex works and that one day it would be nice to build a computer using similar principles (check Jeff Hawkins's work). The fact that I returned to programming SuperMemo in the late evening, i.e. very bad time for creative work, seems to indicate that the passion has not kicked in yet.

Nov 23, 1987 looked identical. I am not sure why I did not have any school obligations on Monday, but this might have saved SuperMemo. On Nov 24, 1987, the excitement kicked in and I worked for 8 hours straight (in the evening again). The program had a simple menu and could add new items to the database.

Nov 25, 1987 was wasted: I had to go to school, I was tired and sleepy. We had excruciatingly boring classes in computer architecture, probably a decade behind the status quo in the west.

Nov 26 was free again and again I was able to catch up with SuperMemo work. The program grew to be 15,400 bytes huge. I concluded the program might be " very usefull" (sic!).

On Nov 27, I added 3 more hours of work after school.

Nov 28 was Saturday and I could add 12 enthusiastic hours of non-stop programming. SuperMemo now looked like almost ready for use.

On Nov 29, Sunday, I voted for economic reforms and democratization in Poland. In the evening, I did not make much progress. I had to prepare an essay for my English class. The essay described the day when I experimented with alcohol one day in 1982. I was a teetotaller, but as a biologist, I concluded I need to know how alcohol affects the mind.

Nov 30 was wasted at school, but we had a nice walk home with Biedalak. We had a long conversation in English about our future. That future was mostly to be about science, probably in the US.

Dec 1-4 were wasted at school again. No time for programming. In a conversation with some Russian professor, I realized that I completely forgot Russian in short 6 years. I used to be proudly fluent! I had to channel my programming time into some boring software for designing electronic circuits. I had to do it to credit a class in electronics. I had a deal with the teacher that I would not attend, just write this piece of software. I did not learn anything and to this day I mourn the waste of time. If I was free, I could have invested this energy in SuperMemo.

Dec 5 was a Saturday. Free from school. Hurray! However, I had to start from wasting 4 hours on some "keycode procedure". In those days, even decoding the key pressed might become a challenge. And then another hour wasted on changing some screen attributes. In addition, I added 6 hours for writing "item editor". This way, I could conveniently edit items in SuperMemo. The effortless things you take for granted today: cursor left, cursor right, delete, up, new line, etc. needed a day of programming back then.

Dec 6 was a lovely Sunday. I spent 7 hours debugging SuperMemo, adding "final drill", etc. The excitement kept growing. In a week, I might start using my new breakthrough speed-learning software.

On Monday, Dec 7, after school, I added a procedure for deleting items.

On Dec 8, while Reagan and Gorbachev signed their nuclear deal, I added a procedure for searching items and displaying some item statistics. SuperMemo "bloated" to 43,800 bytes.

Dec 9 was marred by school and programming for the electronics class.

On Dec 10, I celebrated power cuts at school. Instead of boring classes, I could do some extra programming.

On Dec 11, we had a lovely lecture with one of the biggest brains at school: Prof. Jan Węglarz. He insisted that he could do more in Poland than abroad. This was a powerful message. However, in 2018, his Wikipedia entry says that his two-phase method discovery was ignored, and later duplicated in the west because he opted for publishing in Polish. Węglarz created a formidable team of best operations research brains in Poznan indeed. If I did not sway in the direction of SuperMemo, I would sure come with a begging hat to look for an employment opportunity. In the evening, I added a procedure for inspecting the number of items to review each day (today's Workload).

Dec 12 was a Saturday. I expanded SuperMemo by a pending queue editor, and seemed ready to start learning, however, ...

... on Dec 13, I was hit by a bombshell: " Out of memory". I somehow managed to fix the problem by optimizing the code. The last option I needed to add was for the program to read the date. Yes. That was a big deal hack. Without it, I would need to type in the current date at the start of the work with the program. Finally, at long last, in the afternoon, on Dec 13, 1987, I was able to add my first items to my human biology collection: questions about the autonomic nervous system. By Dec 23, 1987, my combined paper and computer databases included 3795 questions on human biology (of which almost 10% already in SuperMemo). Sadly, I had to remove full repetition histories from SuperMemo on that day. There wasn't enough space on 360K diskettes. Spaced repetition research would need to wait a few more years.

Algorithm SM-2

Here is the description of the algorithm used in SuperMemo 1.0. The description was taken from my Master's Thesis written 2.5 years later (1990). SuperMemo 1.0 was soon replaced by a nicer SuperMemo 2.0 that I could give away to friends at university. There were insignificant updates to the algorithm that was named Algorithm SM-2 after the version of SuperMemo. This means there has never been Algorithm SM-1.

I mastered 1000 questions in biology in the first 8 months. Even better, I memorized exactly 10,000 items of English word pairs in the first 365 days working 40 min/day. This number was used as a benchmark in advertising SuperMemo in its first commercial days. Even today, 40 min is the daily investment recommended to master Advanced English in 4 years (40,000+ items).

To this day, Algorithm SM-2 remains popular and is still used by applications such as Anki, Mnemosyne and more.

1988: Two component of memory

Two-component model of long-term memory lays at the foundation of SuperMemo, and is expressed explicitly in Algorithm SM-17. It differentiates between how stable knowledge is in long term memory storage, and how easy it is to retrieve. This remains a little known and quintessential fact of the theory of learning that one can be fluent and still remember poorly.

Components of long-term memory

I first described the idea of two components of memory in a paper for my computer simulations class on Jan 9, 1988. In the same paper, I concluded that different circuits must be involved in declarative and procedural learning.

If you pause for a minute, the whole idea of two components should be pretty obvious. If you take two items right after a review, one with a short optimum interval and the other with a long optimum interval, the memory status of the two must differ. Both can be recalled perfectly (maximum retrievability) and they also need to differ in how long they can last in memory (different stability). I was surprised I could not find any literature on the subject. However, if the literature has no mention of the existence of the optimum interval in spaced repetition, this seemingly obvious conclusion might be hiding behind another seemingly obvious idea: the progression of increasing interval in optimally spaced review. This is a lovely illustration how human progress is incremental and agonizingly slow. We are notoriously bad at thinking out of the box. The darkest place is under the candlestick. This weakness can be broken with an explosion of communication on the web. I advocate less peer review and more bold hypothesizing. I speak of a fantastic example coming from Robin Clarke's paper in reference to Alzheimer's. Strict peer review is reminiscent of Prussian schooling: in the quest for perfection, we lose our creativity, then humanity, and ultimately the pleasure of life.

When I first presented my ideas to my teacher Dr Katulski on Feb 19, 1988, he was not too impressed, but he gave me a pass for computer simulations credit. Incidentally, a while later, Katulski became one of the first users of SuperMemo 1.0 for DOS.

In my Master's Thesis (1990), I added a slightly more formal proof of the existence of the two components (see next). That part of my thesis remained unnoticed.

In 1994, J. Kowalski wrote in Enter, Poland:

We got to the point where the evolutionary interpretation of memory indicates that it works using the principles of increasing intervals and the spacing effect. Is there any proof for this model of memory apart from the evolutionary speculation? In his Doctoral Dissertation, Wozniak discussed widely molecular aspects of memory and has presented a hypothetical model of changes occurring in the synapse in the process of learning. The novel element presented in the thesis was the distinction between the stability and retrievability of memory traces. This could not be used to support the validity of SuperMemo because of the simple fact that it was SuperMemo itself that laid the groundwork for the hypothesis. However, an increasing molecular evidence seems to coincide with the stability-retrievability model providing, at the same time, support for the correctness of assumptions leading to SuperMemo. In plain terms, retrievability is a property of memory which determines the level of efficiency with which synapses can fire in response to the stimulus, and thus elicit the learned action. The lower the retrievability the less you are likely to recall the correct response to a question. On the other hand, stability reflects the history of earlier repetitions and determines the extent of time in which memory traces can be sustained. The higher the stability of memory, the longer it will take for the retrievability to drop to the zero level, i.e. to the level where memories are permanently lost. According to Wozniak, when we learn something for the first time we experience a slight increase in the stability and retrievability in synapses involved in coding the particular stimulus-response association. In time, retrievability declines rapidly; the phenomenon equivalent to forgetting. At the same time, the stability of memory remains at the approximately same level. However, if we repeat the association before retrievability drops to zero, retrievability regains its initial value, while stability increases to a new level, substantially higher than at primary learning. Before the next repetition takes place, due to increased stability, retrievability decreases at a slower pace, and the inter-repetition interval might be much longer before forgetting takes place. Two other important properties of memory should also be noted: (1) repetitions have no power to increase the stability at times when retrievability is high (spacing effect), (2) upon forgetting, stability declines rapidly

We published our ideas with Drs Janusz Murakowski and Edward Gorzelańczyk in 1995. Murakowski perfected the mathematical proof. Gorzelańczyk fleshed out the molecular model. We have not heard much enthusiasm or feedback from the scientific community. The idea of two components of memory is like wine, the older it gets, the better it tastes. We keep wondering when it will receive a wider recognition. After all, we do not live in Mendel's time to keep a good gem hidden in some obscure archive. There are millions of users of spaced repetition and even if 0.1% got interested in the theory, they would hear of our two components. Today, even the newest algorithm in SuperMemo is based on the two-component model and it works like a charm. Ironically, users tend to flock to simpler solutions where all the mechanics of human memory remain hidden. Even at supermemo.com we make sure we do not scare customers with excess numbers on the screen.

The concept of the two components of memory has parallels to prior research, esp. by Bjork.

In the 1940s, scientists investigated habit strength and response strength as independent components of behavior in rats. Those concepts were later reformulated in Bjork's disuse theory. Herbert Simon seems to have noticed the need for memory stability variable in his paper in 1966. In 1969, Robert Bjork formulated the Strength Paradox: a reverse relationship between the probability of recall and the memory effect of a review. Note that his is a restatement of the spacing effect in terms of the two component model, which is just a short step away from formulating the distinction between the variables of memory. This led to Bjork's New Theory of Disuse (1992) that would distinguish between the storage strength and the retrieval strength. Those are close equivalents of retrievability and stability with a slightly different interpretation of the mechanisms that underlie the distinction. Most strikingly, Bjork believes that when retrievability drops to zero, stable memories are still retained (in our model, stability becomes indeterminate). At the cellular level, Bjork might be right, at least for a while, but practise of SuperMemo shows the power of complete forgetting, while, from the neural point of view, retaining memories in disuse would be highly inefficient independent of their stability. Last but not least, Bjork defines storage strength in terms of connectivity, which is very close to what I believe happens in good students: coherence affects stability.

Why aren't two components of memory entering mainstream research yet? I claim that if human mind tends to be short-sighted, and we all are, by design, the mind of science can be truly strangulated by strenuous duties, publish or perish, battles for grants, hierarchies, conflict of interest, peer review, teaching obligations and even the code of conduct. Memory researchers tend to live in a single dimension of "memory strength". In that dimension, they cannot truly appreciate true dynamics of molecular processes that need to be investigated to crack the problem. Ironically, progress may come from those who tend to work in artificial intelligence or neural network. Prodigious minds of Demis Hassabis or Andreas Knoblauch come up with twin ideas by independent reasoning process, models, and simulations. Biologists will need to listen to the language of mathematics or computer science.

Two component model in Algorithm SM-17

The two component model of long-term memory underlies Algorithm SM-17. The success of Algorithm SM-17 is the ultimate practical proof for the correctness of the model.

A graph of actual changes in the value of the two components of memory provides a conceptual visualization of the evolving memory status:

Due to the fact that real-life application of SuperMemo requires tackling learning material of varying difficulty, the third variable involved in the model is item difficulty (D). Some of the implications of item difficulty have also been discussed in the above article. In particular, the impact of composite memories with subcomponents of different memory stability (S).

For the purpose of the new algorithm we have defined the three components of memory as follows:

• Memory Stability (S) is defined as the inter-repetition interval that produces average recall probability of 0.9 at review time
• Memory Retrievability (R) is defined as the expected probability of recall at any time on the assumption of negatively exponential forgetting of homogenous learning material with the decay constant determined by memory stability (S)
• Item Difficulty (D) is defined as the maximum possible increase in memory stability (S) at review mapped linearly into 0..1 interval with 0 standing for easiest possible items, and 1 standing for highest difficulty in consideration in SuperMemo (the cut off limit currently stands at stability increase 6x less than the maximum possible)

Proof

The actual proof from my Master's Thesis follows. For a better take on this proof, see Murakowski proof.

Proof by Murakowski

Here is an improved proof by Murakowski:

It has been found in earlier research that the optimum spacing of repetitions in paired-associate learning, understood as the spacing which takes a minimum number of repetitions to indefinitely maintain a constant level of knowledge retention (e.g. 95%), can roughly be expressed using the following formulae ( Wozniak and Gorzelańczyk 1994).

• (1) I ₁=C ₁
• (2) I _i=I _i-1*C ₂

where:

• I _i - inter-repetition interval after the i-th repetition
• C ₁ - length of the first interval (dependent on the chosen knowledge retention, and usually equal to several days)
• C ₂ - constant that denotes the increase of inter-repetition intervals in subsequent repetitions (dependent on the chosen knowledge retention, and the difficulty of the remembered item)

The above formulae have been found for human subjects using computer optimization procedures employed to supervise the process of self-paced learning of word-pairs using the active recall drop-out technique. [...]

As it will be shown below, the widely investigated strength of memory (or synaptic potentiation) does not suffice to account for the regular pattern of optimum repetition spacing: [...]

1. We want to determine the set of (molecular) variables involved in storing memory traces that will suffice to account for the optimum spacing of repetitions. Let us, initially, assume two correlates of these variables in learning that is subject to optimum spacing as expressed by Eqns. (1) and (2):

   r - time which remains from the present moment until the end of the current optimum interval (optimum interval is the interval at the end of which the retention drops to the previously defined level, e.g. 95%)

   s - length of the current optimum interval.

2. Just at the onset of the i-th repetition, r=0, while s _i> s _i-1>0 ( s _i denotes s right at the onset of the i-th repetition). This indicates that there is no function g ₁ such that s=g ₁( r), i.e. s cannot be a function of r only.
3. During the inter-repetition interval, r(t ₁)<> r(t ₂) if t ₁<>t ₂ (t denotes time and r(t) denotes r at the moment t). On the other hand, s(t ₁)= s(t ₂) ( s(t) denotes s at the moment t). This shows that there is no function g ₂ such that r=g ₂( s), or we would have: r(t ₁)=g ₂( s(t ₁))=g ₂ ( s(t ₂))= r(t ₂), which leads to a contradiction. r cannot be a function of s only.
4. In Steps 2 and 3 we have shown that r and s are independent, as there are no functions g ₁ and g ₂ such that s=g ₁( r) or r=g ₂( s). This obviously does not mean that there exists no parameter x and functions y _s and y _r such that s=y _s(x) and r=y _r(x).
5. It can be shown that r and s suffice to compute the optimum spacing of repetitions (cf. Eqns. (1) and (2)). Let us first assume that the two following functions f _r and f _s are known in the system involved in memory storage: r _i=f _r( s _i) and s _i=f _s( s _i-1). In our case, these functions have a trivial form f _r: r _i= s _i and f _s: s _i= s _i-1*C ₂ (where C ₂ is the constant from Eqn. (2)). In such a case, the variables r and s are sufficient to represent memory at any moment t in optimum spacing of repetitions. Here is a repetition spacing algorithm which shows this to be true:
   1. assume that the variables r _i and s _i describe the state of memory after the i-th repetition
   2. let there elapse r _i time
   3. let there be a repetition
   4. let the function f _s be used to compute the new value of s _i+1 from s _i
   5. let the function f _r be used to compute the new value of r _i+1 from s _i+1
   6. i:=i+1
   7. goto 2

The above reasoning shows that variables r and s form a sufficient set of independent variables needed to compute the optimum spacing of repetitions. Obviously, using a set of transformation functions of the form r’’=Tr( r’) and s’’=Ts( s’), one can conceive an infinite family of variable pairs r- s that could describe the status of the memory system. A difficult choice remains to choose such a pair r- s that will most conveniently correspond with molecular phenomena occurring at the level of the synapse.

The following terminology and interpretation is proposed by the authors in a memory system involving the existence of the r- s pair of variables: the variable R, retrievability, determines the probability with which a given memory trace can be invoked at a given moment, while the variable S, stability of memory, determines the rate of decline of retrievability as a result of forgetting, and consequently the length of inter-repetition intervals in the optimum spacing of repetitions.

Assuming the negatively exponential decrease of retrievability, and the interpretation of stability as a reciprocal of the retrievability decay constant, we might conveniently represent the relationship between R and S using the following formula (t denotes time):

(3) R=e ^-t/S

The transformation functions from the pair r- s used in Steps 1-5 of the reasoning, to the proposed interpretation R-S will look as follows (assuming the definition of the optimum inter-repetition interval as the interval that produces retention of knowledge K=0.95):

(4) S=- s/ln(K)

(5) R=e ^{-( s- r)/S}

The relationship between the stability after the i-th repetition (S _i) and the constants C ₁ and C ₂ determining the optimum spacing of repetitions as defined by Eqns. (1) and (2) can therefore be written as:

(6) S _i=-(C ₁*C ₂ ^i-1)/ln(K)

and finally, retrievability in the optimum spacing of repetitions can be expressed as:

(7) R _i(t)=exp ^{(t*ln(K)/(C ₁*C ₂ i-1))}

where:

• i - number of the repetition in question
• t - time since the i-th repetition
• R _i(t) - retrievability after the time t passing since the i-th repetition in optimum spacing of repetitions
• C ₁ and C ₂ - constants from Eqns. (1) and (2)
• K - retention of knowledge equal to 0.95 (it is important to notice that the relationship expressed by Eqn. (7) may not be true for retention higher than 0.95 due to the spacing effect resulting from shorter intervals)

Two components of memory in SuperMemo

SuperMemo has always been based on the two component model, which emerged in an increasingly explicit form over time. The constant C ₂ in Eqn. (2) in Murakowski proof above represents stability increase. In 2018, stability increase is represented in SuperMemo as matrix SInc[]. C ₂ says how much inter-repetition intervals should increase in learning to meet the criteria of admissible level of forgetting. In reality, C ₂ is not a constant. It depends on a number of factors. Of these, the most important are:

• item difficulty (D)(see: complexity): the more difficult the remembered piece of information the smaller the C ₂ (i.e. difficult material must be reviewed more often)
• memory stability (S): the more lasting/durable the memory, the smaller the C ₂ value
• probability of recall ( retrievability)(R): the lower the probability of recall, the higher the C ₂ value (i.e. due to the spacing effect, items are remembered better if reviewed with delay)

Due to those multiple dependencies, the precise value of C ₂ is not easily predictable. SuperMemo solves this and similar optimization problems by using multidimensional matrices to represent multi-argument functions and adjusting the value of those matrices on the basis of measurements made during an actual learning process. The initial values of those matrices are derived from a theoretical model or from previous measurements. The actually used values will, over time, differ slightly from those theoretically predicted or those derived from data of previous students.

For example, if the value of C ₂ for a given item of a given difficulty with a given memory status produces an inter-repetition interval that is longer than desired (i.e. producing lower than desired level of recall), the value of C ₂ is reduced accordingly.

Here is the evolution of stability increase (constant C ₂) over years:

• in the paper-and-pencil version of SuperMemo (1985), C ₂ was indeed (almost) a constant. Set at the average of 1.75 (varying from 1.5 to 2.0 for rounding errors and simplicity), it did not consider material difficulty, stability or retrievability of memories, etc.
• in early versions of SuperMemo for DOS (1987), C ₂, named E-Factor, reflected item difficulty for the first time. It was decreased for bad grades and increased for good grades
• SuperMemo 4 (1989) did not use C ₂, but, to compute inter-repetition intervals, it employed optimization matrices for the first time
• in SuperMemo 5 (1990), C ₂, named O-Factor was finally represented as a matrix and it included both the difficulty dimension as well as the stability dimension. Again, entries of the matrix would be subject to the measure-verify-correct cycle that would, starting with the initial value based on prior measurements, produce a convergence towards the value that would satisfy the learning criteria
• in SuperMemo 6 (1991), C ₂, in the form of the O-Factor matrix would be derived from a three-dimensional matrix that would include the retrievability dimension. The important implication of the third dimension was that, for the first time, SuperMemo would make it possible to inspect forgetting curves for different levels of difficulty and memory stability
• in SuperMemo 8 (1997) through SuperMemo 16, the representation of C ₂ would not change much, however, the algorithm used to produce a quick and stable transition from the theoretical to the real set of data would gradually get more and more complex. Most importantly, new SuperMemos make a better use of the retrievability dimension of C ₂. Thus, independent of the spacing effect, the student can depart from the initial learning criteria, e.g. to cram before an exam, without introducing noise into the optimization procedure
• in SuperMemo 17 (2016), C ₂ finally took the form based on the original two-component model. It is taken from stability increase matrix (SInc) that has three dimensions that represent the three variables that determine the increase in stability: complexity, stability and retrievability. The SInc matrix is filled up with data during learning using a complex algorithm known as Algorithm SM-17. The stability increase matrix can be inspected in SuperMemo 17 with Tools : Memory : 4D Graphs ( Stability tab)

1989: SuperMemo adapts to user memory

Introducing flexible interval function

SuperMemo 2 was great. Its simple algorithm has survived in various mutations to this day in popular apps such as Anki or Mnemosyne. However, the algorithm was dumb in the sense that there was no way of modifying the function of optimum intervals. The findings of 1985 were set in stone. Memory complexity and stability increase were expressed by the same single number: E-factor. It is a bit like using a single lever in a bike to change gears and the direction of driving.

Individual items could adapt the spacing of review by changes to their estimated difficulty. Those changes could compensate for errors in the function of optimum intervals. Even if the algorithm was slow to converge on the optimum, in theory, it was convergent. The main flaw was that, in Algorithm SM-2, new items would not benefit from the experience of old items.

Algorithm SM-4 was the first attempt to arm SuperMemo with universal adaptability. It was completed in February 1989. In the end, adaptability was too slow to show up, but inspiration gathered with Algorithm SM-4 was essential for further progress, esp. in understanding the problem of stability-vs-accuracy in spaced repetition. In short, Algorithm SM-4 was too stable to be accurate. This was quickly remedied in Algorithm SM-5 just 7 months later. Here is an excerpt from my Master's Thesis to explain the details:

Rigid SuperMemo 4

It did not take me long to realize that the verification-correction cycle in the new algorithm was too long. It was not much different than running the 1985 eperiment on the computer. To determine decade-long intervals, I needed a decade to pass to test the outcomes of the review. This led to Algorithm SM-5 seven months later. Here is the problem with Algorithm SM-4 as described in my Master's Thesis:

Remnants of SuperMemo 4 in new SuperMemos

Interestingly, you can still see the matrix of optimum intervals in newer versions of SuperMemo. The matrix is not used by the algorithm, however, it is displayed in SuperMemo statistics as it informs the user about the impact of complexity on the prospects of items in the learning process.

If you compare a matrix produced by SuperMemo 5 in 8 months of use, you will notice significant similarity to a matrix produced in two decades of use of Algorithm SM-8:

Algorithm SM-4

Here is the outline of Algorithm SM-4 as described in my Master's Thesis:

Problems with interval matrix

In addition to slow convergence, Algorithm SM-4 showed that the use of matrix of intervals leads to several problems that could easily be solved by replacing intervals with O-factors. Those additional flaws led to a fast implementation of Algorithm SM-5 yet in 1989. Here is a short analysis that explained the flaws in the use of the matrix of optimum intervals:

SuperMemo 5

The idea of the matrix of optimum intervals was born on Feb 11, 1989. On Mar 1, 1989, I started using SuperMemo 4 to learn Esperanto. Very early I noticed that the idea needs revision. The convergence of the algorithm was excruciatingly slow.

By May 5, I had a new idea in my mind. This was, in essence, the birth of stability increase function, except, in the optimum review of SuperMemo there would be no retrievability dimension. The new algorithm would use the matrix of optimum factors. It would remember how much intervals need to increase depending on memory complexity and current memory stability. Slow convergence of Algorithm SM-4 also inspired the need to randomize intervals (May 20).

In the meantime, progress had to be delayed because of school obligations again. With Krzysztof Biedalak, we decided to write a program for developing school tests that could be used with SuperMemo. The project was again used to wriggle out from other obligations in classes with open-minded Dr Katulski who has become a supporter of all-things SuperMemo by now.

I spent summer on practical training in the Netherlands when progress was slow due to various obligations. One of the chief reasons for slowness was extreme dieting that resulted from the need to save money to pay my PC 1512 debts. I also hoped to earn something extra to buy a hard disk for my computer. All my work was possible thanks to the courtesy of Peter Klijn of the University of Eindhoven. He just gave me his PC for my private use for the whole period of stay. He did not want my good work over SuperMemo to slow down. It was the first time I could keep all my files on a hard disk and it felt like a move from an old bike to Tesla Model S.

Only on Oct 16, 1989, the new Algorithm SM-5 was completed and I started using SuperMemo 5. I remarked in my notes: " A great revolution is in the offing". The progress was tremendous indeed.

I had a couple of users of SuperMemo 2 that were ready to upgrade to SuperMemo 5. I demanded only one price: they will start with the matrix of optimum factors initialized to a specific value. This was to validate the algorithm and make sure it carries no preconceived prejudice. All preset optimization matrices would converge nicely and fast. This fact was then used to claim universal convergence and the algorithm was described as such in the first-ever publication about spaced repetition algorithms.

Algorithm SM-5

SuperMemo uses a simple principle: "use, verify, and correct". After a repetition, a new interval is computed with the help of the OF matrix. The "relevant entry" to compute the interval depends on the repetition (category) and item difficulty. After the interval elapses, SuperMemo calls for the next repetition. The grade is used to tell SuperMemo how well the interval "performed". If the grade is low, we have reasons to believe that the interval is too long and the OF matrix entry is too high. In such cases, we slightly reduce the OF entry. The relevant entry here is the one that was previously used in computing the interval (i.e. before the interval started). In other words, it is the entry that is (1) used to compute the interval (after n-th repetition) and then (2) used to correct the OF matrix (after the n+1 repetition).

Here is an outline of Algorithm SM-5 as presented in my Master's Thesis:

In accordance with the previous observations, the entries of the OF matrix were not allowed to drop below 1.2. In Algorithm SM-5, by definition, intervals cannot get shorter in subsequent repetitions. Intervals are at least 1.2 times as long as their predecessors. Changing the E-Factor category increases the next applied interval only as many times as it is required by the corresponding entry of the OF matrix.

Criticism of Algorithm SM-5

Anki manual includes a passage that is surprisingly critical of Algorithm SM-5 (Apr 2018). The words are particularly surprising as Algorithm SM-5 has never been published in full (the version above is just a rough outline). Despite the fact that the words of criticism were clearly uttered in goodwill, they hint at the possibility that if Algorithm SM-2 was superior over Algorithm SM-5, perhaps it is also superior over Algorithm SM-17. If that was the case, I would have wasted the last 30 years of research and programming. To this day, Wikipedia "criticises" "SM3+". "SM3+" is a label first used in Anki manual that has been used at dozens of sites on the web (esp. those that prefer to stick with the older algorithm for its simplicity). A comparison between Algorithm SM-2 and Algorithm SM-17 is presented here.

Erroneous claim in Anki manual

My Master's Thesis published in excerpts in 1998 at supermemo.com included only a rough description of the Algorithm SM-5. For the sake of clarity, dozens of minor procedures were not published. Those procedures would require a lot of tinkering to ensure good convergence, stability, and accuracy. This type of tinkering requires months of learning combined with analysis. There has never been a ready out-of-the box version of Algorithm SM-5.

The source code of Algorithm SM-5 has never been published or opened, and the original algorithm could only be tested by users of SuperMemo 5, in MS DOS. SuperMemo 5 became freeware in 1993. It is important to notice that random dispersal of intervals around the optimum value was essential for establishing convergence. Without the dispersal, the progression of the algorithm would be agonizingly slow. Similarly, matrix smoothing was necessary for consistent behavior independent of the richness of data collected for different levels of stability and item complexity.

Multiple evaluations done in 1989, and since, have pointed to an unquestionable superiority of Algorithm SM-5 and later algorithms over Algorithm SM-2 in any metric studied. Algorithm SM-17 might actually be used to measure the efficiency of Algorithm SM-5 if we had volunteers to re-implement that ancient code for use with our universal metrics. We have done this for Algorithm SM-2 thus far for the implementation cost was insignificant. Needless to say, Algorithm SM-2 lags well behind in its predictive powers, esp. for suboptimum levels of retrievability.

Even a basic understanding of the underlying model should make it clear that a good implementation would yield dramatic benefits. SuperMemo 5 would adapt its function of intervals to the user's memory. SuperMemo 2 was set in stone. I am very proud that wild guesses made in 1985 and 1987 stood the test of time, but no algorithm should trust the judgment of a humble student with 2 years experience in the implementation of spaced repetition algorithms. Instead, SuperMemo 4 and all successive implementations made fewer and fewer guesses and provided better and faster adaptability. Of all those implementations, only SuperMemo 4 was slow to adapt and was replaced in 7 months by a superior implementation.

There is no ill-will in Anki criticism but I would not be surprised if the author pushed for an implementation and speedy move to self-learning rather than spending time on tinkering with procedures that did not seem to work as he expected. In contrast, back in 1989, I knew Algorithm SM-2 was flawed, I knew Algorithm SM-5 was superior, and I would spare no time and effort in making sure the new concept was perfected to its maximum theoretical potential.

Excerpt from the Anki manual (April 2018):

Anki was originally based on the SuperMemo SM5 algorithm. However, Anki's default behaviour of revealing the next interval before answering a card revealed some fundamental problems with the SM5 algorithm. The key difference between SM2 and later revisions of the algorithm is this:

• SM2 uses your performance on a card to determine the next time to schedule that card
• SM3+ use your performance on a card to determine the next time to schedule that card, and similar cards

The latter approach promises to choose more accurate intervals by factoring in not just a single card's performance, but the performance as a group. If you are very consistent in your studies and all cards are of a very similar difficulty, this approach can work quite well. However, once inconsistencies are introduced into the equation (cards of varying difficulty, not studying at the same time every day), SM3+ is more prone to incorrect guesses at the next interval - resulting in cards being scheduled too often or too far in the future.

Furthermore, as SM3+ dynamically adjusts the "optimum factors" table, a situation can often arise where answering "hard" on a card can result in a longer interval than answering "easy" would give. The next times are hidden from you in SuperMemo so the user is never aware of this.

After evaluating the alternatives, the Anki author decided that near-optimum intervals yielded by an SM2 derivative are better than trying to obtain optimum intervals at the risk of incorrect guesses. An SM2 approach is predictable and intuitive to end users, whereas an SM3+ approach hides the details from the user and requires users to trust the system (even when the system may make mistakes in the scheduling).

Some details for anyone who cares:

• the fact that SuperMemo 5 used past performance of all items to maximize performance on new items is an advantage, not a "problem". Even more, that is the key to the power of adaptability
• inconsistent grading has been a problem for all algorithms. On average, adaptability helps find out the average effect of misuse, esp. if inconsistencies are consistent (i.e. user keep committing similar offences in similar circumstances)
• mixed difficulties are handled by SuperMemo 5 much better because while both difficulty and stability increase are coded by E-factor in SuperMemo 2, in SuperMemo 5 and later, those two properties of memory are separated
• interval predictions have been proven superior in SuperMemo 5 and the claim "more prone to incorrect guesses" can only be explained by errors in implementation
• lower grades could bring longer intervals if matrix smoothing is not implemented. That part of the algorithm has only been described verbally in my thesis
• intervals and repetition dates have always been prominently displayed in SuperMemo, even in most simpler implementations (e.g. for handheld devices, smartphones, etc.). Nothing is hidden from the user. Most of all, forgetting index and burden statistics are in full view and make it possible to see if SuperMemo keeps its retention promise and at what cost to workload

Of all the above claims, only one might be true. SuperMemo 2 might indeed be more intuitive. This problem has plagued SuperMemo for years. Each version is more complex, and it is hard to hide some of that complexity from users. We will keep trying though.

Our official response published at supermemopedia.com in 2011 seems pretty accurate today:

Evaluation of SuperMemo 5 (1989)

SuperMemo 5 was so obviously superior that I did not collect much data to prove my point. I made only a few comparisons for my Master's Thesis and they left no doubt.

Theoretic proof of superiority of newer algorithms

Anki's criticism of SuperMemo 5 calls for a simple proof in the light of modern spaced repetition theory. We can show that today's model of memory can be mapped onto the models underlying both algorithms: Algorithm SM-2 and Algorithm SM-5, and the key difference between the two is the missing adaptability of the function of optimal intervals (represented in Algorithm SM-5 as Matrix OF).

Let SInc=f(C,S,R) be a stability increase function that takes complexity C, stability S, and retrievability R as arguments. This function determines the progressive increase in review intervals in optimum learning.

Both Algorithms, SM-2 and SM-5 ignore the retrievability dimension. In theory, if both algorithms worked perfectly, we could assume they aim at R=0.9. As it can be measured in SuperMemo, both algorithms fail at that effort for they do not know relevant forgetting curves. They simply do not collect forgetting curve data. This function was introduced only in Algorithm SM-6 in 1991.

However, if we assume that 1985 and 1987 heuristics were perfect guesses, in theory, the algorithm could use SInc=F(C,S) with constant R of 90%.

Due to the fact that SM-2 uses the same number, EF for stability increase and for item complexity, for SM-2 we have SInc=f(C,S) equation represented by EF=f'(EF,interval), where it can be shown easily with data that f<>f'. Amazingly, the heuristic used in SM-2 made this function work by decoupling the actual link between the EF and item complexity. As data shows that SInc keeps decreasing with S, this means that in Algorithm SM-2, by definition, all items would need to gain on complexity with each review if EF was to represent item complexity. In practical terms, Algorithm SM-2 uses EF=f'(EF,interval), which translates to SInc(n)=f(SInc(n-1),interval).

Let us assume that the EF=f(EF,interval) heuristic was an excellent guess as claimed by users of Algorithm SM-2. Let SInc be represented by O-factor in Algorithm SM-5. We might then represent SInc=f(C,S) as OF=f(EF,interval).

For Algorithm SM-2, OF would be constant and equal to EF, in Algorithm SM-5, OF is adaptable and can be modified depending on algorithm's performance. It seems pretty obvious that penalizing the algorithm for bad performance by a drop to OF matrix entries and rewarding it by an increase in OF entries is superior to keeping OF constant.

On a funny twist, as much as supporters of Algorithm SM-2 claim it performs great, supporters of neural network SuperMemo kept accusing algebraic algorithms of: lack of adaptability. In reality, the adaptability of Algorithm SM-17 is best to-date as it is based on the most accurate model of memory.

It is conceivable that heuristics used in SM-2 were so accurate that the original guess on OF=f(EF,interval) needed no modification. However, as it has been shown in practical application, the matrix OF quickly evolves and converges onto values described in Wozniak, Gorzelańczyk 1994. They differ substantively from the assumption wired into Algorithm SM-2.

Summary:

• today: SInc=f(C,S,R), 3 variables, f is adaptable
• sm5: SInc=f(C,S,0.9), 2 variables, f is adaptable
• sm2: SInc=f(SInc,S,0.9) - 1 variable, f is fixed

Convergence

Algorithm SM-5 showed fast convergence, which was quickly demonstrated by users who began with univalent OF matrices. It was quite a contrast with Algorithm SM-4.

Matrix smoothing

Some of the criticism from the author of Anki might have been a result of not employing matrix smoothing that was an important component of the algorithm and is still used in Algorithm SM-17.

Random dispersal of intervals

One of the key improvements in Algorithm SM-5 was random dispersal of intervals. On one hand, it dramatically accelerated the optimization process, on the other, it caused a great deal of confusion in users of nearly all future versions of SuperMemo: "why do the same items with the same grade use a different interval at each try?". Minor deviations are precious. This was laid bare when "naked" Algorithm SM-17 was released in early SuperMemo 17. It could be seen that users who keep a lot of leeches in their collections would easily hit "local minima" from which they could never get out. The random dispersion was restored with some delay. The period of "nakedness" was needed for accurate observations of the algorithm, esp. in multi-decade learning process like my own.

1990: Universal formula for memory

Optimum review vs. intermittent review

By 1990, I had no doubt. I had a major discovery at hands. I cracked the problem of forgetting. I knew the optimum timing of review for simple memories. Once I secured the permission to describe my findings in my Master's Thesis, my appetite for discovery kept growing. I hoped I might find a universal formula for long-term memory. A formula that would help me track the behavior of memory for any pattern of exposure or retrieval.

I already had a collection of data that might help me find the formula. Before discovering the optimum spacing of repetitions in 1985, I used pages of questions for review of knowledge. The review was chaotic and determined by the availability of time, the need, or the mood. I called that " intermittent learning". I had recall data for individual pages and for each review. That was the ideal kind of data that did not have the periodicity of SuperMemo. The exact kind of data needed to solve the problem of memory. However, I had that data on paper only.

In Spring 1990, I recruited my sister to do the typing. No. I do not have a younger sister who would do that eagerly. My sister was 17 years my senior. Being a bit inconsiderate for her time, I used her love to make her do the donkey work. I feel guilty about it. She died just two years later. I never had a chance to repay her contribution to the theory of spaced repetition, which she never even had a chance to understand. Starting on May 1, 1990, she used my time away from the computer to transfer the data from paper to the computer. It took her many days of slow typing. It was worth it.

Model of intermittent learning

Throughout the summer of 1990, instead of focusing on my Master's Thesis, I worked on the " model of intermittent learning". It was not unusual for me to work for 10 hours straight, or go to sleep at 7 am empty-handed, or leave the computer churning the numbers overnight.

Persistence and tinkering pay. Only teens can afford it and should be given the space and the freedom. Despite being 28 years old, I was being tolerated at home pretty well. Like an immature teen. I lived at my sister's apartment where I could leech on her kindness. Long hours at the computer were excused as " working on my Master's Thesis". The truth was nobody asked me to do it, nobody demanded it, it did not even push SuperMemo much ahead. It was a sheer case of scientific curiosity. I just wanted to know how memory works.

I had dozens of pages of questions and their repetition history. I tried to predict "memory lapses per page". I used root-mean-square deviation for lapse prediction (denoted below as Dev