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Lord of the rings: the Spire numerical towers
Denis Rosset, researcher in quantum physics
Friday, June 14, 2019 · 1:47 p.m. · 31m 33s
Spire defines around 80 typeclasses, including 30 coming from algebra and cats-kernel. We’ll see how much of that structure is dictated by mathematical laws, and which parts are the result of design decisions that balance different tradeoffs. In particular, we’ll discuss the different roles played by typeclasses in the Scala ecosystem: as encoding operations obeying well-defined laws, as enabling the use of a particular syntax for those operations, if possible close to the mathematical notation of a domain (and subfields often disagree on the notation!), defining a context in which a combination of typeclasses implicitly imposes additional laws (for example, the ordering of numbers and addition), enabling the user to change the variant of a relation being used (Order), singling out one variant of a structure as canonical (cats: the additive Group for integers), as selecting a particular algorithm for an operation (integer factorization: deterministic or Monte-Carlo).
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so i get the the name is a play on what's us by others and all the
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computers numbers and mathematician came was absorption that on on us rains so the rings will come
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a few times during the presentation but just think a structure that as addition and multiplication uh every time you see
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real uh so there's a ways to to contact me people giving me money and what they do in my spare time
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it by cutting yeah but they don't pay me to to work on like a pencil stuff um
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i would start by thinking like people contributed to those project in even started
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this project 'cause i'm not the the mean old office by um just like
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pushing it for whilst um and and the the be like lots of contributions
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um what are we what what we are going to talk about this type classes from limerick i think
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um just just a quick check check we used a type classes in all ways fun yeah we start classes in this room
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i would have probably even a very different talk like you say um but right right now i can uh
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move it move really fast on that um soon like delegates but i'm still using the the alternate energy sometimes
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and two example that i'll will be part of the um of of spy and the
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lever is it's it's based on is all the l. that's enable you to compare things
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but often when you have a um a type just like all the it's it's specialise is something
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that's already exist for example all the us ah more specific than possible all those was sometime the
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the operation can fail and possible though is going to be more specific then equality will be just compare
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stuff uh if they are they are equal um that's already in cats can and what a spy up
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put on top of things to to compute with numbers like a edition plus so that would be like another pipe less
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ah but we already have some type this is understandably so why why spire why don't we just use
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never you're really a flat not been map and it is good enough um the thing is that there's there's another
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boy did you you just enough so that you can do things like sequence that some sequences salted and nothing more
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um for example he uh uh you see that the the the the can of
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the inherent insulation in distance library yeah we're because everything that's new michael can be compared
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but you look at for example complex numbers of polynomial as you can add them
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you can multiply them but they are not all the on the real line so
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you would not be able to use all type classes fall more general
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application that just i'm still optional type that represented in the summer library
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um and they ah actually interesting things they put them somewhere while i'm not
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and that means that if you turn the car there is something like um
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the bottle of these uh we absorption or something that or tricky functional like
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it can it can afford channel so it's it's not very low for the the the on all else that has object
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to my knowledge in in the celebrity um so like them i put like one those those
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white box so that other type classes the operation it again by adding that that's implicit in school
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uh now we move to the type level towers in like that like zero it's um most who quite
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um and even comprise some of the things all uh the meets john
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let this isn't logic is like a family is is complete wall instead
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and i put in for this is all the possible rings it's another wall instead of um
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that's been uh i mean it on this on the slide because
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if you jump in and start to just file this is what to do will uh in
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your face when you start so egg possible the all the it's a room cats can and
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i will probably if i determine elevates sign the it one on top of the all the your notion of a zero and on
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which side of the zero you you are um and truncated division
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will speak about it in a minute it's a very interesting story
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then you have the single structure that's what enables you to
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for example concatenate strings with the same operation that you would
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use to combine numbers mean it by adding them not um and they are like the this is family of things like
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community logic it basically where it would be an operation that work on bit strings
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for example so this is very specialised i'm i'm not going to spend time on that
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um what's interesting for those operation that some apportion we commute for example two plus three street was too
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but no if i use your version of we've got a neat thing straying is a link between a plus the string be
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it's going to be able to try reverse the argument then is be a it's it's not the same thing
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so we have those lions and then on the right uh it's exactly the same thing as it would melinda something group
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a little bit more into details later um but sometime you want to use i did innovation
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only to be getting notation so all the things that that three ticket edits by
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and some people complain but maybe when we should come with a new
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way to do these things just putting up there to pause to discuss
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um on top of the addition and multiplication you have a series of adobe structure that
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tell you all additional a summer's vacation and the violence ah uh in that stuff is basically
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can you compute investors from education for example was into girls you can't
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compute the inverse of two uh one how does it's not and they get
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um but maybe if you have a field you can compute in verses so this was basically
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the textbook mathematics with a few exceptions like for example in mass you a unique vocalisation domains
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we should be in in that how well like in the middle but we for some
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reason we put them aside and that's a design choice i would put speaker with data
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uh okay eh i wanted to show that's just uh we have
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most of them the skull every oh yeah it's uh the doubles conflicted
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which is an interesting name i'm surprised wanted it's first we are going to law to look at what's the value of
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the classes like young salad is that what people use by uh we are type was that one can just add things
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but but you know you have laws and you get this for free and that's kind of the value that that we provide
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um second one uh i want to talk about is the variance in korea runs the the fact
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we can create a new type classes i just with i think things and what does that mean for oh i can system
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pointer is going to be which which instances do we provide possible
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types one for a syntax i hinted that that's like uh this reputation
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although we closer that is when we have like so many violence and then we somehow need to
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i've were combination of type classes away level uh and finally or um
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sometimes even a math i'm i'm i think boxing to say that thing exists
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and is unique but when you compute with it you have different ways of
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getting the results maybe when you compute stuff you you have more freedom there
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to the big thing then then when you work with uh id on mathematical things um so that's not was last what's
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um and what i'd like to do like this this all stuff i'm i want to go repossessed collateral
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um so that's going to be that uh the thing is that as 'cause something's really bad like if you look at the the printed
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uh the scary part defines normal so intake war blah blah blah blah blah uh
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and somehow you will i that's it's it's it depends a euclidean domain why not
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uh oh and then somehow you have this notion of zero and two was negative infinity it so much on the phone all the
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so implicitly they bring the notion of an although into that thing and you don't with not all ugly and the men's all the object
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what it is and that was the problems by that we saw them you know we're better than has got so many different you know
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escalate stuff from from us if um and that person's was the person out there itself
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it was fun that going on the right side um it's always like
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you divide something by something in what remains after you've you've done your division
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um so for example seventy three times to plus one that's not very controversial that's
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uh i should get minus seven you have two ways of defining it like
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minus all times two plus one or maybe a three times to minus one
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and when it does menu module things always annoying that you get a negative result so ingrained
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to remain the the this division because for example let's say i want to have a ring buffet
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and say i want to do things module one second of sound that and i have offsets
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and and every time i something negative gonna get a negative upset in that hour
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it's very annoying to put a while loop every every time i need to do that
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um something i want to use the definition that would have a a plus sign up here but you know exactly what what gets
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i mean the java level it's very interesting if used the doesn't operate only get the sign of
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the uh the don'ts and if used in a meeting together knob good design of the the diesel
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so it's not internally consistent it's more of a if you sporting points maybe what you
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should say that i can't about seven by to see which one time two point five
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plus the oh sorry that's one half time seven or this is wrong um
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uh i don't know that would that should be trip on five ten two plus you uh wanted to avoid
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yep but that what what i want to say remembers you can always divide so they should me no remainder
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but sometime you get actually when you use the person of our car
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you get mine's one or plus one so you know what's this um
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it's something that we didn't have been inspired much at the time but you
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can use also like this this modern remainder stuff to divide putting on the outs
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so in high school i did like this kind of long division was putting
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announced and then you get a a the remainder of the division should be five
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and pauses by we happen no males we want that classes that enable us to do
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the right thing when people use pronouns on one of the skull ah a question that
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so to actually the story you you there's a very nice paper i'm i'm across of research uh they look at it um
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and what you have is that's you divide it be divided by it's more the um should you should get an indigo
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questions and and you you big the what you divide the
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and she did the times you the questions plus some remained
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that's not controversial and then like the the rest should be smaller than the the the the diesel
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what's more i was like this this double lines on the certainly the absolute value so now we have two interpretations
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the first one is about okay the division basically i'm going to divide the trend line by the the blue line and i
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get the remainder swing thing 'cause i can fit three times the blue thing the right thing and uh uh this when it's
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something to write like this is true time blue plus was but um but you have to
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to do that i need to uh notion of a comparing lengths when in the order to that's
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and the other notion is this faulty and division and then you say it's very ups like to say like this thing is duty and function
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um which matches the division the the sort of definition fall back into gets
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so this is why file didn't see the problem until and until much later
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um but you don't in all the for example for polynomial that's uh like all begin and it's the renewal degree
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if a extra for three or two or x. on on uh
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i think the biggest one and that's my that's my it can function
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i'm i'm going to skip over the agenda division because it's not very
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it's it's not something we all share um the trunk of the division it's
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so you you and you have the thing was the the next it's a very nice picture but when you
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have signs uh you have a bit of freedom you you can decide all exactly you you kind of signs
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and this is by the way i'm researching that paper of the of the book that's what programming languages like all
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do they handle that thing and no inspired out what it means we have actually two versions okay that of role models
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and it is uh like the result would follow the sign of the they didn't all these
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so now when i want to index in in re module blah and melts
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it can be negative i just use of mother and i don't need to
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a bad okay so what i like about this story that's it's not
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others very abstract question what's uh you can division and truncated division it
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why should i care and in the end we provide users a better story because now when your index something and
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i basically loops back you have a primitive that's the sound and a super usable
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so that's the first that lecture uh what's you know following that low approach when brings you
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a second story that's like violence all we've uh that
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a few times from from martin during the holidays in today
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that's uh we don't want global coherence uh implicit delegates we visit a local um
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it's good that people for example can reverse and all the and when the
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cool uh the salt to this matter than on on the collection the can
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give us a things if if they need they need to um
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the problem is that you can you can basically have many all those in in in your library in which one
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you using is not going to be checked by the compiler it's more like the documentation or
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um like the the basically the culture you are in your project that will which to do that um
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so for example you need to be supportive for that if you have started said that
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you always using the same although maybe you can but it's not going to check that
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um and sometime although is going to be use in combination with sort of
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things you can just work at the division we are also using and all the
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but it needs to be compatible with not this multiple up actresses and at the same time
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um so that was my story four point two that we we have to be without a second problem
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by the way i'm not going to get answered doing that work more like always issues in it like
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what to type classes people get into heated debates online that this is the one
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way to do it uh i don't think so and and the other the the results
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um the so when when we have to make decisions it's gonna get instances um
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so we have uh some hope alright in cats cannot uh tells you that's a
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it's a sets and you can combine stuff and it's a sensitive so you can basically
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pound is is what uh where where you want it means you can just get rid of bounces which is really cool to weeds things
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um it's also cool for people with a bottle computing because you can just get your sequence in
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whatever chance and you compute done in power and and then you can bring things back
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together and it's saying so that's that's the value of the type that's actually the um
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intake uh addition would not be the only operation that respect that multiplication of
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the walks ah and it's also should leave and like mean in maxed uh something
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so when you in in cats when you do somebody asked for some group of ins what do you do what do you what you did
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um and this is your draw like it depends what are you doing like data processing you walk in mathematics
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as an example for example an electrical system you're going to speak
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what faces and all you compute phases um if you specify them at
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an angle you just add the angles module trend in sixty degrees
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but if they are complex numbers of menus one you multiply them which
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which one would you pick when it's um as an example so cats but it's
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a uh it's the number of talkative and given the opinion in north america and
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they have a map that is like only talked some somebody day uh i know
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if i combine them this is doing the right thing it's adding i'm like well maybe
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which i was looking at something like that to get my might maximum salary of all my carrier so that's
00:15:19
often in countries like retirement benefits are tied to the high salary which is
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why suddenly like people in this last last year frontline wanted to so many
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additional responsibilities but because then being you have a nice bonus
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um which it you would use the mass operation not not some
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and it's a would look like in switzerland is the thing that's healthcare so expensive that the states of the size it and it'll get your revenue
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maybe you'll get the revenue over the past three years the minimum one not going
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to add things uh so no no no this is a problem because all do we
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um i got the fact that's it's going to be a different something group uh every time
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when we divide them on read of map string beans uh i don't know the answer to this
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um false point and actually quite cool because like twenty minutes in switch and big
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buttons and i stand discussion uh so it's it's it when you when you bring
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it that class in into your project not only you bring like the structure and
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the roles which also going some syntax um and yeah there is something really different between
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walking was cut out walking which fourteen languages and weeding mathematical of physics papers because you can do this
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stop the paper and you defined annotation if it's something you can
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see that's using slightly different notation as value for you on paper
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as as i suppose you have to binary operation you didn't them like blind lab here people
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put little circles on one so it's not just on a class like you can't mandate something different
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um and it's okay matt because you're complex is always a paper and when other
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people's papers together you uh it's the research i was to read stuff in right
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bring bring them in the same space but we when you what programs you
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need to compose and you need to shadow notation between the the different pieces um
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i'm not familiar with liz at all but have a that is something
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that people basically we defined the wall every time to start a project um
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it's got a and when you discuss like cats is going to to get you
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to tell you what to do with uh the type classes like what's the syntax
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um and it means that in sky we're gonna apparently
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we have the vacation so if we get groups fall
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like abstract hoops so that's uh that's kennel and what's
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pile does is basically the same thing just with different names
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oh by the way it's gonna z. mixes all these notations level i think this is something oh i like
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type of alex's in those those much better so like that's combine in in gets gonna uh it's this kind of
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squared plus operation it was was it they you have an anti adamant that basically does nothing when when combined
00:18:03
and every element as any of us are not to be hooked group every animal
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doesn't invest that gives you a when combined with the or something did you and
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um but you get the same thing with was into goes and in addition uh
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addition is a subjective you have a a natural element which is zero and
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every numbers a negative number when you add them you you get is you
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something communication assertive one is unusual elements and
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then everyone of us a reciprocal uh an
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amount not your this is kind of partial thing that we do it it will but
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acts upon zero um this brings you to one so we have exactly the same
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structure the same type class uh like in in in an idea of like twenty min
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uh but because we are going to use them full two things um we're going to have
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a different swatches for that um and kind of the convention when noise that's the the something group from cats
00:19:05
is going to useful monumental things combining strains maps and
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we are going to use this notation for the market applications
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um and with both of them actually because we knew him
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when you we're what computation you're going to add and multiply things so we are going to have
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type classes than marriage um a plus and and and and time sensation combining them
00:19:30
and you we will gets additional laws like this repetitive um which i've written on on
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the bottom of the slide so when there was a nice way out of this um
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i think we just have to live with the fact that we are
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going to love triplicate structures interpret it allows entropy get uh that's infrastructure
00:19:50
but luckily that this really is a lever is when you're the user you just change
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the name of the scene using objectionable didn't bought for the syntax and and it's that um
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i don't i wish is gonna add more obstruction to combine in in you know that's
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i want to i i have a type a and i want to deform good structures on it
00:20:10
one was that petition one was this all the station i think it would make things very confusing
00:20:15
some but i don't know if this is like the the the end of the stock that's that don't fit of things
00:20:22
um ah a control things i've seen that happen all the time like when when i got was built
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like standards is a little less lasting go to what's the alley cats like this kind of you know
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kind of pogo tori a a a modern if um i want
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to say here that's in that complex no less classes that okay
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yeah okay if they become if they contribute in laws when
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they are using commission was all the things so for example
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uh all the i was there something yeah yeah then it's happening um
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quite doubly i this problem when i i define
00:20:58
um elaborate to learn algebra announced aspect thousands best matrices
00:21:03
and the idea of that that laborious to use as little
00:21:07
stuff is as possible so just to construct as bass matrix
00:21:11
i don't i don't like what i was i think i just want to do something at the on mon and t.
00:21:16
so this is all my idea like something group and and one night structure would look like something good would tell you
00:21:21
all to comment things and and he would tell you like this is this is the empty object and i can check if
00:21:28
actually i've also like a way to compare voice quality i can check if that's and
00:21:33
and i can be like my uh matrix contraction things like feel tabulate and and so on
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and then the mother it would combine something good an empty and you know anti ad right to exist because it's then part of something a bit
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um but i think that was the end of their part i'm not something really really
00:21:53
and fun it's gonna it's when you're like the phones capabilities for your type and you want to create like no other
00:22:02
that type class at the top of the unit returns a that is a for a recognition it becomes extremely messy
00:22:08
so here and it's it's not contrived example this is what we have right now we have this list of things
00:22:14
on the left equality possible all the all the and like do you do we have a sign and so on
00:22:19
um another white we are like what are the capabilities of phone numbers i don't know
00:22:24
for example and sign in together as they don't have a notion of uh
00:22:27
taking the negative negates we don't have that so so we have like things
00:22:33
that leave but if we step of this lead on the right and then
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if we want to close that iraqi we basically need to take the the
00:22:40
we're pairs of uh left and and right and that would do a little two hundred two hundred
00:22:46
classes all traits in in the rubbery we we we want to that um
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the problems that we uh uh we have most of awful type class managing like five instance of that and
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that and they are cory run then okay you can construct that's bigger thing and it gives you additional uh operations
00:23:04
um i don't even know with all that would look like the
00:23:08
only kind of struggles to jewel operation we have is in returns
00:23:13
the thing is not too bad because we can use convention here and a kind of a dogfight thought
00:23:18
well so that's uh the title of the talking um and just closer that is inside those those tell us
00:23:25
so for example on the left uh we have the all during power and i choose that to get the division would
00:23:30
lead on on that power and then you of all the things related to plus and and and multiplication and they don't overlap
00:23:37
okay when you ask an instance of begin together we'll get something on from the latin something from the right that you don't have a commissioner
00:23:45
and that that keeps your some sanity that crazy too
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um yeah just wanted to mention that uh when we test lows that's it so
00:23:54
that's where you see that actually to get a division leaves in in both tell us
00:23:59
um but we we made the choice that's in the test you
00:24:02
require to get the division but also a competitive read so it's like
00:24:06
a pleasant times what you don't have negative um and then but um
00:24:13
so this is all we we do it that's when you're going to use it you're going to bring balls and animal from boston was at the same time
00:24:19
this is something that sonic and it installs got nobody's going to check that this is going
00:24:23
and and seen a because the compiler doesn't all but so that's uh the trade up here
00:24:29
uh let's thing and and then lost i mean or whatever for minutes remaining um
00:24:35
some things were allowed to students like i give them the text book on whatever maps we are going to start it and then it on the computer and
00:24:41
often math textbook tell you something exist and it's it's it's fine with um but when you need to go
00:24:47
on the computer you need to pick up that that's what show an algorithm that's wanted sometime they don't exist or
00:24:52
um it says it's a very nice problematical you are actually do computation
00:24:58
yeah want to think about unique factories asian domains that are like it's
00:25:02
it's a it's something that leaves at the very bottom of the kind of
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rain last time sorry yeah working i just means that's your
00:25:10
whole concept uh for units of for most numbers plus minus one
00:25:13
and you have like blind animals and you can take every element of your set and you can do vocalisation and it's huge
00:25:20
um and if you take all to do that on the j. v. m. probably can together you don't have something that i mean you think of this
00:25:27
so you you you put some seventy parameters and i was going to say you
00:25:31
know like uh and it's probably prime oh i'm so it can be fact out that's
00:25:38
it is i like all much uh said that you want to have this is a guy wanted to give you should put seventy didn't have then i mean
00:25:45
in the end you did that probability of this thing for me is going
00:25:48
to be low well then uh gamma ray fitting bits in your memories will
00:25:53
in the an article it's good enough but um some peabody mass would
00:25:57
not approve computer prove that are based on on such a arguments so
00:26:02
uh maybe you want and and here we have a problem we can provide the top class uh because
00:26:07
you would have to pick a certainty parameter and maybe between different domains like you would have
00:26:13
you have it that it takes a day multiple could but you ought malls that idea i don't know if i'm
00:26:17
so here i think the right choices to let the user choose with impulse liked what algorithm they want to use
00:26:24
uh at that that means that for for this unit factories is from the main we can put that in
00:26:28
the in ring tones uh your key we have this
00:26:31
as to the outside and then you bought whatever you need
00:26:36
so with that i rather my conclusion think i'm quite good with i mean yeah it's good um so
00:26:44
like the like the sixth in points when designing a like um an idea in in a in a p.
00:26:50
uh within that's basically having laws is is not controversial um we have
00:26:56
a problem was with cory once and and violence of like classes but
00:27:00
this can be probably remitted by medication like all too and buttons like oh to use that properly
00:27:05
uh we should get something better chemical instances this is control your to pick
00:27:09
the one that makes more sense for your use of is maybe that's open
00:27:14
a syntax i think he has brought up it's is unavoidable but if you need
00:27:18
to you have to make a choice is considering like we're going to use your library
00:27:23
clothing that is easy use a towel pattern and i presume section this is not very controversial but
00:27:30
then we rely that's just sort of whatever is not the sleek and for what's in the text
00:27:35
so that was like the short summary of this uh why did they make this talk because um meetings by
00:27:42
uh and now we need to move to just got three it's a good excuse to break or the things
00:27:47
um and really agree with you would be to to be the mainstream language that has the most will just
00:27:52
and michael features and i i think we can be we already better than ask al um and that that's
00:27:58
gonna be like the story of scanner that that we have that and everybody can use its uh that's that's
00:28:14
any questions
00:28:16
of course we um
00:28:22
or
00:28:25
so yeah how to see and how do you envision these what do you think is going to be some
00:28:32
jen affect my jeep around my ear and stuff in k. s. one project yeah um i
00:28:39
think one of for the user extensively that that we have cats canada and type travellers abroad inspire
00:28:45
um
00:28:47
but it was say that i want to move things back in into cats i want to be sure that the yeah work
00:28:52
you we have is is is good i think it's some but
00:28:55
maybe it's a bit too precise that i i don't know um
00:29:00
but i think we should probably decide on time line and and try to get like
00:29:04
a like an answer that that line in in in get things to be point agree that's
00:29:08
um i don't discuss with this before working on cats all the while yet about this like um
00:29:16
ah we have time for more questions
00:29:19
we will
00:29:22
um
00:29:25
alright and think oh no week right
00:29:30
is that the one on the right answer i'm i'm i'm curious um i guess i guess something about how
00:29:37
figure out if we need to shift things around between three per among the three projects would be
00:29:44
um do we have any idea how mom like what's slices of these libraries people using
00:29:51
i'm i'm just kind of curious 'cause like cats cats goes farther than scholars it
00:29:55
for instance uh_huh uh_huh in terms there's just has a cartels it stops with with more
00:30:01
yeah can excuse to has groups lattices yeah in cats colonel yeah in algebra goes for him
00:30:08
really idea people using
00:30:11
and what i'm using in an hour working like a conventional physics it's very different from like what people would use
00:30:18
goth kids fall so i try to get a feeling doings colour they like a what what are the structures then
00:30:23
working with like very nice things like uh why why this book but
00:30:26
the math in their mathematical switch off of accounting classes for you know uh
00:30:33
computing tax for example it's it's uh but they have no idea how or what's what's what on for business um
00:30:40
thing why no i would kind of stay around one person you
00:30:44
mogul computing on the market analysis like you were going to crazy man
00:30:48
to make up for shows like put that that would be the the baseline me maybe what's in a car computer graphics text book for example
00:30:56
um and then the question is how much position do you want in the intermediate steps because it's very
00:31:01
difficult to to choose home things back in a hindrance
00:31:05
yeah actually when when when you're not a controlling that's
00:31:10
um but why no we're super precise and i don't know much that's uh that's about it
00:31:15
i mean it's probably discussion we're not going to have in our remaining minutes or something
00:31:20
but any any people interested please come to me at the next wake up draft audio input on that's
Conference Program
Keynote: Some Mistakes We Made When Designing Implicits (And Some Things We Got Right)
Martin Odersky, Professor EPFL, Chairman Typesafe
June 14, 2019 · 9:13 a.m.
303 views
Brave New World - tales of PureScript and Haskell in production
Felix Mulder, Snoop Dogg
June 14, 2019 · 10:22 a.m.
151 views
Actors Design Patterns and Arrowised FRP
June 14, 2019 · 10:53 a.m.
Taking Resources to the Type Level
June 14, 2019 · 11:48 a.m.
Lord of the rings: the Spire numerical towers
Denis Rosset, researcher in quantum physics
June 14, 2019 · 1:47 p.m.
Formal verification of Scala programs with Stainless
June 14, 2019 · 2:20 p.m.
118 views
Exploring Scala Tooling ecosystem
Jeferson David Ossa, Colombia
June 14, 2019 · 3:12 p.m.
TwoFace values: a bridge between terms and types
June 14, 2019 · 3:43 p.m.
GADTs in Dotty
June 14, 2019 · 4:41 p.m.
302 views
Closing Remarks
June 14, 2019 · 5:49 p.m.
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