Relational Algebra with Fancy Types

Last time, I tried to give a primer of relations and relational algebra using the Haskell type type Rel a b = [(a,b)]. In this post we’re going to look at these ideas from a slightly different angle. Instead of encoding relations using value level sets, we’ll encode relations in the type system. The Algebra of Programming Agda repo and the papers quoted therein are very relevant, so if you’re comfortable wading into those waters, give them a look. You can find my repo for fiddling here

At this point, depending on what you’ve seen before, you’re either thinking “Yeah, sure. That’s a thing.” or you’re thinking “How and why the hell would you do such a ridiculous thing.”

Most of this post will be about how, so let’s address why first:

1. Examining relations in this style illuminates some constructions that appear around the Haskell ecosystem, particularly some peculiar fellows in the profunctor package.
2. More syntactic approach to relations allows discussion of larger/infinite domains. The finite enumerations of the previous post is nice for simplicity, but it seems you can’t get far that way.
3. Mostly because we can – It’s a fun game. Maybe a useful one? TBD.

With that out of the way, let’s go on to how.

We will be using some Haskell extensions in this post, at the very least GADTs and DataKinds. For an introduction to GADTs and DataKinds, check out this blog post. DataKinds is an extension that reflects every data constructor of data types to a type constructor. Because there are values True and False there are corresponding types created'True and 'False. GADTs is an extension of the type definition mechanism of standard Haskell. They allow you to declare refined types for the constructors of your data and they infer those refined type when you pattern match out of the data as well, such that the whole process is kind of information preserving.

We will use the GADT extension to define relational datatypes with the kind

. That way it has a slot a for the “input” and b for the “output” of the relation. What will goes in these type slots will be DataKind lifted types like 'True, not ordinary Haskell types like Int. This is a divergence from from the uses of similar kinds you see in Category, Profunctor, or Arrow. We’re doing a more typelevel flavored thing than you’ll see in those libraries. What we’re doing is clearly a close brother of the singleton approach to dependently typed programming in Haskell.

Some examples are in order for what I mean. Here are two simple boolean functions, not and and defined in ordinary Haskell functions, and their equivalent GADT relation data type.

You can already start to see how mechanical the correspondence between the ordinary function definition and our new fancy relation type. A function is often defined via cases. Each case corresponds to a new constructor of the relation and each pattern that occurs in that case is the pattern that appears in the GADT. Multiple arguments to the relations are encoded by uncurrying everything by default.

Any function calls that occur on the right hand side of a function definition becomes fields in the constructor of our relation. This includes recursive calls and external function calls. Here are some examples with a Peano style natural number data type.

We can also define things that aren’t functions. Relations are a larger class of things than functions are, which is part of their utility. Here is a “less than equal” relation LTE.

You can show that elements are in a particular relation by finding a value of that relational type. Is ([4,7], 11) in the relation Plus? Yes, and I can show it with with the value PS (PS (PS (PS PZ))) :: Plus (4,7) 11 . This is very much the Curry-Howard correspondence. The type R a b corresponds to the proposition/question is $(a,b) \in R$ .

The Fun Stuff : Relational Combinators

While you need to build some primitive relations using new data type definitions, others can be built using relational combinators. If you avoid defining too many primitive relations like the above and build them out of combinators, you expose a rich high level manipulation algebra. Otherwise you are stuck in the pattern matching dreck. We are traveling down the same road we did in the previous post, so look there for less confusing explanations of the relational underpinnings of these constructions, or better yet some of the references below.

Higher order relational operators take in a type parameters of kind

and produce new types of a similar kind. The types appearing in these combinators is the AST of our relational algebra language.

The first two combinators of interest is the composition operator and the identity relation. An element $(a,c)$ is in $R \cdot Q$ if there exists a $b$ such that $(a,b) \in R$ and $(b,c) \in Q$. The fairly direct translation of this into a type is

The type of the composition is the same as that of Profunctor composition found in the profunctors package.

Alongside a composition operator, it is a knee jerk to look for an identity relation and we do have one

This is also a familiar friend. The identity relation in this language is the Equality type.

We can build an algebra for handling product and sum types by defining the appropriate relational combinators. These are very similar to the combinators in the Control.Arrow package.

The converse of relations is very interesting operation and is the point where relations really differ from functions. Inverting a function is tough. Conversing a relation always works. This data type has no analog in profunctor to my knowledge and probably shouldn’t.

Relations do not have a notion of currying. The closest thing they have is

Lattice Operators

For my purposes, lattices are descriptions of sets that trade away descriptive power for efficiency. So most operations you’d perform on sets have an analog in the lattice structure, but it isn’t a perfect matching and you’re forced into approximation. It is nice to have the way you perform these approximation be principled, so that you can know at the end of your analysis whether you’ve actually really shown anything or not about the actual sets in question.

The top relation holds all values. This is represented by making no conditions on the type parameters. They are completely phantom.

Bottom is a relation with no inhabitants.

The meet is basically the intersection of the relations, the join is basically the union.

A Lattice has an order on it. This order is given by relational inclusion. This is the same as the :-> combinator can be found in the profunctors package.

Relational equality can be written as back and forth inclusion, a natural isomorphism between the relations. There is also an interesting indirect form.

Relational Division

If we consider the equation (r <<< p) :-> q with p and q given, in what sense is there a solution for r? By analogy, this looks rather like r*p = q, so we’re asking a kind of division question. Well, unfortunately, this equation may not necessarily have a solution (neither do linear algebraic equations for that matter), but we can ask for the best under approximation instead. This is the operation of relational division. It also appears in the profunctor package as the right Kan Extension. You’ll also find the universal property of the right division under the name curryRan and uncurryRan in that module.

One formulation of Galois connections can be found in the adjunctions file. Galois Connections are very slick, but I’m running out of steam, so let’s leave that one for another day.

Properties and Proofs

We can prove many properties about these relational operations. Here a a random smattering that we showed using quickcheck last time.

Odds and Ends

• Recursion Schemes – Recursion schemes are a methodology to talk about recursion in a point free style and where the rubber meets the road in the algebra of programming. Here is an excellent series of articles about them. Here is a sample of how I think they go:
• Higher Order Relations?
• Examples of use. Check out the examples folder in the AoP Agda repo. These are probably translatable into Haskell.
• Interfacing with Singletons. Singletonized functions are a specialized case or relations. Something like?
• A comment to help avoid confusion. What we’ve done here feels confusingly similar to profunctor, but it is in fact distinct I think. Profunctors are described as a categorical generalization of relations , but to be honest, I kind of don’t get it. Despite many of our constructions appearing in the profunctor package, the profunctor typeclass itself appears to not play a role in our formulation. There just isn’t a good way to dimap under our relations as written, unless you construct free profunctors. Converse at the least is a wrench in the works.
• Star and graphs. Transition relations are a powerful methodology. A transition relation is in some respects the analog of a square matrix. We can iteratively compose it with itself.

Notes on Getting Started in OCaml

I know a bit of Haskell. It’s the functional programming language I have the strongest background in. OCaml is very similar to Haskell, which is why I haven’t felt a strong need to learn it. Having had to delve into it for necessity because of work I think that was an oversight. The biggest thing for me is being able to more easily read a new set of literature and see familiar things in a new light, which is very refreshing.

Getting OCaml

opam is the package manager. Follow the instructions to install it and get your environment variables setup. It’ll tell you some extra commands you have to run to do so. You use it to install packages via opam install packagename. You can also use it to switch between different ocaml compiler versions via command like opam switch 4.08.1.

Dune is a build tool. You can place a small config file called dune in your folder and it can figure out how to appropriately call the compiler. Dune is in flux, so check out the documentation. What I write here may be wrong.

Here’s an example execution. Note that even though the file is called main.ml in this example, you call build with main.exe. And exec requires the ./ for some reason. Weird.

Here’s a dune file with some junk in it. You make executables with blocks. You include a list of the files without the .ml suffix require by the executable in the modules line. You list libraries needed in the libraries line.

You want to also install merlin. opam install merlin. Merlin is a very slick IDE tool with autocomplete and type information. dune will setup a .merlin file for you

The ReasonML plugin is good for vscode. Search for it on the marketplace. It is the one for OCaml also. ReasonML is a syntactic facelift intended for the web btw. I don’t particularly recommend it to start. There are also emacs and vim modes if that is what you’re into.

The enhanced repl is called utop. Install it with opam install utop. Basic repl usage: Every line has to end with ;;. That’s how you get stuff to be run. Commands start with #. For example #quit;; is how you end the session. #use "myfile.ml";; will load a file you’ve made. Sometimes when you start you need to run #use "topfind";; which loads a package finder. You can load libraries via the require command like #require "Core";;

#help;; for more.

A Quick Look at the Language

With any new language I like to check out Learn X from Y if one is available.

https://learnxinyminutes.com/docs/ocaml/

Here are some shortish cheat sheets with a quick overview of syntax

https://github.com/alhassy/OCamlCheatSheet

https://ocaml.org/docs/cheat_sheets.html

More In Depth Looks

This is a phenomenal book online book built for a Cornell course: https://www.cs.cornell.edu/courses/cs3110/2019sp/textbook/

Real World OCaml is also quite good but denser. Very useful as a reference for usage of Core and other important libraries.

The reference manual is also surprisingly readable https://caml.inria.fr/pub/docs/manual-ocaml/ . The first 100 or so pages are a straightforward introduction to the language.

https://github.com/janestreet/learn-ocaml-workshop Pretty basic workshop. Could be useful getting you up and running though.

Useful libraries

Core – a standard library replacement. Becoming increasingly common https://github.com/janestreet/core It is quite a bit more confusing for a newcomer than the standard library IMO. And the way they have formatted their docs is awful.

Owl – a numerical library. Similar to Numpy in many ways. https://ocaml.xyz/ These tutorials are clutch https://github.com/owlbarn/owl/wiki

Bap – Binary Analysis Platform. Neato stuff

Lwt – https://github.com/ocsigen/lwt asynchronous programming

Graphics – gives you some toy and not toy stuff. Lets you draw lines and circles and get keyboard events in a simple way.

OCamlGraph – a graph library

Jupyter Notebook – Kind of neat. I’ve got one working on binder. Check it out here. https://github.com/philzook58/ocaml_binder

Menhir and OCamlLex. Useful for lexer and parser generators. Check out the ocaml book for more

fmt – for pretty printing

Interesting Other Stuff (A Descent into Lazy Writing)

https://discuss.ocaml.org/ – The discourse. Friendly people. They don’t bite. Ask questions.

https://github.com/ocaml-community/awesome-ocaml Awesome-Ocaml list. A huge dump of interesting libraries and resources.

An excerpt of cool stuff:

• Coq – Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.
• Why3 – Why3 is a platform for deductive program verification. It provides a rich language for specification and programming, called WhyML, and relies on external theorem provers, both automated and interactive, to discharge verification conditions.
• Alt-Ergo – Alt-Ergo is an open-source SMT solver dedicated to the proof of mathematical formulas generated in the context of program verification.

http://ocamlverse.github.io/ – A pretty good set of beginner advice and articles. Seems like I have a lot of accidental overlap. Would’ve been nice to find earlier

https://www.cl.cam.ac.uk/teaching/1617/L28/materials.html – advanced functional programming course. Interesting material.

TAPL – https://www.cis.upenn.edu/~bcpierce/tapl/ Has implementations in OCaml of different lambda calculi. Good book too.

It is not uncommon to use a preprocessor in OCaml for some useful features. There is monad syntax, list comprehensions, deriving and more available as ppx extensions.

https://whitequark.org/blog/2014/04/16/a-guide-to-extension-points-in-ocaml/ ppx perepsorcssor. ocamlp4 5 are both preprocessors too

https://tarides.com/blog/2019-05-09-an-introduction-to-ocaml-ppx-ecosystem.html

https://blog.janestreet.com/archive/ The jane street blog. They are very prominent users of ocaml.

https://opensource.janestreet.com/standards/ Jane Street style guide

Oleg Kiselyov half works in Haskell, half in OCaml, so that’s cool.

Oleg metaocaml book. MetaOCaml is super cool. http://okmij.org/ftp/ML/MetaOCaml.html And the switch funtionality of opam makes it easy to install!

Oleg tagless final http://okmij.org/ftp/tagless-final/index.html

https://github.com/ocamllabs/higher

Cohttp, LWT and Async

https://github.com/backtracking/ocamlgraph ocaml graphs

https://mirage.io/ Mirage os. What the hell is this?

https://github.com/ocamllabs/fomega

https://github.com/janestreet/hardcaml

ppx_let monaidc let bindings

some of the awesome derivinig capabilites are given by ppx_jane. SExp seems to be a realy good one. It’s where generic printing is?

dune build lambda.bc.js will make a javascript file. That’s pretty cool. Uses js_of_ocaml. The js_of_ocaml docs are intimidating https://ocsigen.org/js_of_ocaml/dev/manual/overview

http://ocsigen.org/js_of_ocaml/dev/api/

Note you need to install both the js_of_ocaml-compiler AND the library js_of_ocaml and also the js_of_ocaml-ppx.

Go digging through your _build folder and you can find a completely mangled incomprehensible file jsboy.bc.js. But you can indeed import and use it like so.

dune build --profile release lambda.bc.js putting it in the release profile makes an insane difference in build size. 10mb -> 100kb

There is also bucklescript for compiling to javascript. Outputs readable javascript. Old compiler fork?

Check out J.T. Paach’s snippets. Helpful

Dune:

https://gist.github.com/jtpaasch/ce364f62e283d654f8316922ceeb96db

Z3 ocaml

https://gist.github.com/jtpaasch/3a93a9e1bcf9cae86e9e7f7d3484734b

https://jobjo.github.io/2019/04/24/ocaml-has-some-new-shiny-syntax.html

#require “ppx_jane”;; in utop in order to import a thing using ppx

And argument could be made for working from a docker

Weird dsls that generate parsers and lexers. Also oddly stateful.

Took a bit of fiddling to figure out how to get dune to do.

Otherwise pretty straight forward encoding

expereince rport: using f omega as a teaching language

Because they aren’t hidden behind a monadic interface (for better or for worse), OCaml has a lot more of imperative feel. You could program in a subset of the language and have it not feel all that different from Java or python or something. There are for loops and while loops, objects and classes, and mutable variables if you so choose. I feel like the community is trying to shy away from these features for most purposes however, sitcking to the functional core.

However, it does let you do for loops and has an interesting community anddifferent areas of strength.

Maybe more importantly it let’s you access a new set of literature and books. Sligthly different but familiar ideas

I think Core is bewildering for a newcomer.

Rando Trash Poor Style OCaml Snippets

lex_lisp.mll : simplistic usage of ocamllex and menhir

parse_lisp.mly

Doinking around with some graphics

A couple Advent of code 2018

A little Owl usage

Doing Basic Ass Shit in Haskell

Haskell has lots of fancy weird corners, but you want to get rippin’ and runnin’

The Haskell phrase book is a new useful thingy. Nice and terse.

https://typeclasses.com/phrasebook

This one is also quite good https://lotz84.github.io/haskellbyexample/

I also like what FP complete is up to. Solid set of useful stuff, although a bit more emphasis towards their solutions than is common https://haskell.fpcomplete.com/learn

I was fiddling with making some examples for my friends a while ago, but I think the above do a similar better job.

https://github.com/philzook58/basic-ass-shit

Highlights include:

Makin a json request

Showing a plot of a sine function

Doing a least squares fit of some randomly created data

I love Power Serious. https://www.cs.dartmouth.edu/~doug/powser.html Infinite power series using the power of laziness in something like 20 lines

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42.8903&rep=rep1&type=pdf Jerzy Karczmarczuk doing automatic differentiation in Haskell before it was cool. Check out Conal Elliott’s stuff after.

Very simple symbolic differentiation example. When I saw this in SICP for the first time, I crapped my pants.

https://www.cs.kent.ac.uk/people/staff/dat/miranda/whyfp90.pdf Why functional Programming Matters by John Hughes

https://www.cs.cmu.edu/~crary/819-f09/Backus78.pdf John Backus emphasizing escaping the imperative mindset in his 1978 Turing Award speech. A call to arms of functional programming

https://www.cs.tufts.edu/~nr/cs257/archive/richard-bird/sudoku.pdf Richard Bird defining sudoku solutions and then using equation reasoning to build a more efficient solver

Here’s how I find useful Haskell packages:

I google. I go to hackage (if I’m in a subpage, click on “contents” in the upper right hand corner). Click on a category that seems reasonable (like “web” or something) and then sort by Downloads (DL). This at least tells me what is popular-ish. I look for tutorials if I can find them. Sometimes there is a very useful getting started snippet in the main subfile itself. Some packages are overwhelming, others aren’t.

The Real World Haskell book is kind of intimidating although a lovely resource.

The wiki has a pretty rockin set of tutorials. Has some kind soul been improving it?

I forgot learn you a Haskell has a chapter on basic io

When you’re ready to sit down with Haskell more, the best intro is currently the Haskell Book

You may also be interested in https://www.edx.org/course/introduction-functional-programming-delftx-fp101x-0 this MOOC

https://github.com/data61/fp-course or this Data61 course

Then there is a fun infinitude of things to learn after that.

______

More ideas for simple examples?

This post is intentionally terse.

IO is total infective poison.

standard output io

mutation & loops. You probably don’t want these. They are not idiomatic Haskell, and you may be losing out on some of the best lessons Haskell has to offer.

file IO

web requests

http://www.serpentine.com/wreq/tutorial.html

web serving – scotty

image processing

basic data structures

command line arguments

plotting

Parallelism and Concurrency

CAV 2019 Notes: Probably Nothin Interestin’ for You. A bit of noodling with Liquid Haskell

I went to the opening workshops of CAV 2019 in New York this year (on my own dime mind you!) after getting back from joining Ben on the long trail for a bit. The tutorials on Rosette and Liquid Haskell were really fun. Very interesting technology. Also got some good ramen and mochi pudding, so it’s all good. Big Gay Ice Cream was dece.

Day 1 workshops

Calin Belta http://sites.bu.edu/hyness/calin/.Has a new book. Control of Temporal logic systems. Automata. Optimized. Partition space into abstraction. Bisimulation https://www.springer.com/gp/book/9783319507620

Control Lyapunov Function (CLF) – guarantees you are going where you want to go

Control Barrier Function – Somehow controls regions you don’t want to go to.

Lyapunov function based trajectory optimization. You somehow have (Ames 2014) http://ames.gatech.edu/CLF_QP_ACC_final.pdf Is this it?

Differential flatness , input output linearization

Temproal logic with

Rise of Temporal Logic

Linear Temporal Logic vs CTL

Fixpoint logic,

Buchi automata – visit accepting state infinite times

equivalency to first order logic

monadic logic, propositions only take 1 agrument. Decidable. Lowenheim. Quantifier elimination. Bounded Mondel property

Languages: ForSpec, SVA, LDL, PSL, Sugar

method of tableau

Polytopic regions. Can push forward the dynmaics around a trajectory and the polytope that you lie in. RRT/LQR polytopic tree. pick random poitn. Run.

Evauating branching heuristics

branch and prune icp. dreal.

branch and prune. Take set. Propagate constraints until none fire.

branching heuristics on variables

largest first, smearing, lookahead. Try different options see who has the most pruning. Non clear that helped that muhc

QF_NRA. dreal benchmarks. flyspeck, control, robotics, SMT-lib

http://capd.sourceforge.net/capdDynSys/docs/html/index.html

Rosette

verify – find an input on which the assertions fail. exists x. not safe

debug – Minimal unsat core if you give an unsat query. x=42/\ safe(s,P(x))\$ we know thia is unsat because of previous step

solve – exists v si.t safe(v)

synthesis – exists e forall x safe(x,P(x))

define-symbolic, assert, verify, debug, solve, sythesis

Rosette. Alloy is also connected to her. Z Method. Is related to relational logic?

https://homes.cs.washington.edu/~emina/media/cav19-tutorial/index.html

http://emina.github.io/rosette/

Building solver aided programming tool.

symbolic compiler. reduce program all possible paths to a constraint

Cling – symbolic execution engine for llvm

implement intepreter in rosette

Symbolic virtual machine

layering of languages. DSL. library (shallow) embedding. interpreter (deep) embedding.

deep embedding for sythesis.

I can extract coq to rosette?

how does it work?

reverse and filter keeping only positive queries.

symbolic execution vs bounded model checking

symbolic checks every possible branch of the program. Cost is expoentntial

CBMC.

type driven state merging. Merge instances of primitiv types. (like BMC), value types structurally ()

instance Merge Int, Bool, Real — collect up SMT context

vs. Traversable f => Merge (f c) – do using Traversable

symbolic union a set of guarded values with diskoint guard.

merging union. at most one of any shape. bounded by number of possible shapes.

puts some branching in rosette and some branch (on primitives) in SMT.

symbolic propfiling. Repair the encdoing.

tools people have built.

strategy generation. That’s interesting. builds good rewrite rules.

serval.

certikso komodo keystone. fintie programss

IS rosette going to be useful for my work? coooooould be

https://ranjitjhala.github.io/

another thing we could do is galois connections between refinements. Pos, Zero, Neg <-> Int

Liquid Haskell uses SMT solvers to resolve it’s type checking requirements.

Agda et al also work very much via unification. Unification is a broad term but it’s true.

It also has a horn clause solver for inference. Every language needs some kind of inference or you’d go insane. Also it is piggybacking on haskell

It’s not as magical as I thought? Like seeing the magicians trick. It really does understand haskell code. Like it isn’t interpretting it. When it knows facts about how (+) works, that is because the refined type was put in by hand in the prelude connecting it to SMT facts. What is imported by liquid haskell?

The typing environment is clutch. You need to realize what variables are in scope and what their types are, because that is all the SMT can use to push through type checking requirements.

Installing the stack build worked for me. It takes a while . I couldn’t get cabal install to work, because I am not l33t.

Uninterpeted functions. Unmatchability?

It wouldn’t be haskell without a bunch of compiler directives. It is somewhat difficult to find in a single cohesive place what all the syntax, and directives are from liquid haskell. Poking around it best.

• ple
• reflection
• no-termination
• higherorder – what is this?

https://github.com/ucsd-progsys/230-wi19-web course notes

https://github.com/ucsd-progsys/liquid-sf some of software foundations

https://nikivazou.github.io/publications.html niki vazou’s pubs. Check out refinement reflection

https://arxiv.org/pdf/1701.03320 intro to liquid haskell. Interesting to a see a different author’s take

http://goto.ucsd.edu/~nvazou/presentations/ presentations. They are fairly similar to one another.

Liquid haskell gives us the ability to put types on stuff that wasn’t possible before.

Linearity :: f :: {a -> b | f (s ^* a) == s ^* (f a) }

Pullback. {(a,b) | f a == g b}

Equalizer

Many things in categoruy theory rely on the exists unique. Do we have functiona extensionality in Liquid haskell?

product : {(a,b) | f q = x, g q = y, => }

Pushing the boundaries on what liquid haskell can do sounds fun.

Equalizer. The eqaulizer seems prominents in sheaves. Pre-sheaves are basically functors. Sheaves require extra conditions. Restriction maps have to work? Open covers seem important

type Equalizer f g a b = {(e :: a , eq :: a -> b) | f (eq e) = g (eq e) }

I think both the type a and eq are special. e is like an explcit parametrization.

type Eq f g a = {e :: a | f e = g e} I think this is more in the spirit. Use f and g both as measures.

presheaf is functor. But then sheaf is functor that

(a, Eq (F a) (G a)). typelevel equalizer? All types a that F and G agree on.

https://ncatlab.org/nlab/show/equalizer

Records are sheaves – Jon Sterling. Records have subtyping. This gives you a toplogy feeling thing.

{foo | a} {bar | a} -> intersection = {foo bar | b} can inhabit either

union is
or do you want closed records? union is union of fields. intersection is intersection of fields.

In this case a cover would be a set of records with possibly overlapping fields whose combined labels cover the whle space we want to talk about. consistency condition of sheaf/equalizer is that overlapping records fields have to match. I guess {  q.foo = r.foo } ?There is a way to combine all the stuff up. This is exactly what Ghrist was getting at with tables. Tables with shared columns.

data R1 = R1 {foo :: Int, bar :: Int}

{ (r1 :: R1, r2 :: R2) | (foo r1) = (foo r2) } — we manitain duplicates across records.

{. }

if you have a “cover” {foo bar |} {bar fred} {gary larry} whose in

https://www.sciencedirect.com/science/article/pii/S1571066108005264

Sheaves. As a model of concurrency? Gaguen paper.

sheaves as constraint satisfcation? sheafifcation. Constraint solving as a way of fusing the local constraints to be globally consistent.

sheaves as records

sheaves as data fusion

Escardo. Compact data types are those finitely searchable

Continuous funcitons are ~computable? Productive?

http://www.paultaylor.eu/

http://www.paultaylor.eu/ASD/foufct/

http://www.paultaylor.eu/~pt/prafm/

typed recursion theory toplogy

typed computatabiltity theory

Topological notions in computation. Dictionary of terms realted decidable, searchable, semi decidablee

cs.ioc.ee/ewscs/2012/escardo/slides.pdf

https://en.wikipedia.org/wiki/Computable_topology

Through NDArray overloading, a significant fragment of numpy code is probably verifiable.

Need to inspect function annotations to know how to build input type.

@verify() tag

If statements are branching. We are again approaching inspecting functions via probing. But what if we lazily probe. At every __bool__ point, we run a z3 program to determine if there is an avaiable bool left possible (we don’t need to inspect dead code regions. Also would be cool to mention it is a dead region). Curious. We’re reflecting via Z3.

Loops present a problem. Fixed loops are fine. but what about loops that depend on the execution? for i in range(n). I guess again we can hack it…? Maybe. range only takes an integer. we don’t have overload access.

Maybe we need to go into a full eval loop. utterly deconstructing the function and evaluating it statelemnt by statement.

(compare :: a -> a -> Comparison). We could select a choice based on if there is a new one avaialable. Requires access to external store. We lose the thread. How can we know a choice was made? How can we know what the choice was? Did it ask var1 or var2? We can probably do it in python via access to a global store. But in haskell?

while loops take invariant annotations.

It would be cool to have a program that takes

pre conditions. Post ocnditions, but then also a Parameter keyword to declare const variables as deriveable. exists parameter. forall x precondition x => post condition.

Parameter could be of a type to take a dsl of reasonable computations. Perhaps with complexity predicates. and then interpretting the parameter defines the computation.

Or simpler case is parameter is an integer. a magic number.

A Short Skinny on Relations & the Algebra of Programming

I’ve been reading about the Algebra of Programming lately and lovin’ it. See J.N. Oliveira’s draft text in particular and the links in the references. I’ve started exploring the stuff from this post and more over here: https://github.com/philzook58/rel

Why and What?

Relations can expand the power of functional programming for the purpose of specification.

The point of a specification is to be able to write down in a very compact and clear way your intent for a program, more clearly and compactly than a full implementation could be written. It therefore makes sense to add to your specification language constructs that are not necessarily executable or efficient for the sake of compactness and clarity. When one needs executability or efficiency, one writes an implementation whose behavior you can connect to the spec via a formal or informal process.

Functional programming, with it’s focus on the abstraction of the mathematical function, is a good trade-off between executability, efficiency, and expressibility. It lies in a reasonable location between the ideas amenable to reasoning by a human mind and the command-driven requirements of the machines.

Functions are a specialization of relations. Relations extend the mathematical notion of functions with constructs like nondeterministic choice, failure and converse. These constructs are not always obviously executable or efficient. However, they greatly extend the abilities of reasoning and the clarity of expression of a specification.

The point-free style of reasoning about functions extends to a point-free style reasoning about relations, which is known as relation algebra. There are rich analogies with databases, category theory, linear algebra, and other topics.

Plus, I think it is very neato for some reason. If anyone ever thinks something is really neato, isn’t it worth giving it a listen?

A Simple Representation of Relations in Haskell

The simplest description of relations is as a set of tuples. So first let’s talk a bit about the options for sets in Haskell.

There are a couple different reasonable ways to represent sets in Haskell.

• [a] or Vector a
• a -> Bool
• Set a — a tree based Set from the containers package.

These have different performance characteristics and different power. The list [a] is very simple and has specialized pleasant syntax available. The indicator function a -> Bool gives you no ability to produce values of type a, but can easily denote very sophisticated spaces. Set a is a good general purpose data structure with fast lookup. You might also choose to mix and match combinations of these. Interconversion is often possible, but expensive. This is not a complete list of possibilities for sets, for example you may want a representation with a stronger possibility for search.

We can directly use the definition of relations as a set of tuples with the above

But we also have the option to “curry” our relation representations, sort of mixing and matching properties of these representations.

You might also choose to package up multiples of these representations, choosing the most appropriate as the situation requires, see for example the relation package, whose type holds both Map a (Set b) and Map b (Set a).

Despite fiendishly poor performance, for simplicity and list comprehension syntax we are going to be using type Rel a b = [(a,b)] for the remainder of the post.

I’m also taking the restriction that we’re working in bounded enumerable spaces for ease. I assume such a requirement can be lifted for many purposes, but finite spaces like these are especially well tamed. The following typeclass and definition is very useful in this case.

Functions and Relations

Functions can be thought of as relations with the special property that for each left part of the tuple, there is exactly one right side and every possible left side appears. The relation corresponding to a function $f$ looks like $F = \{(x,y) | x \in X, y \in Y, y = f (x)\}$.

There is a natural and slightly clever lifting of function composition to relations. We now check whether there exists a value that is in the right side of one and the left side of the other.

Because of these two operations (and their properties of associativity and absorption), FinRel is a category. We do however need the Eq b restriction to make Rel an instance of the category typeclass, so it does not quite fit the definition of category in base. It is a constrained category.

We can lift the common arrow/categorical combinators up to relations for example.

With these combinators, you have access to many functions on basic non-recursive algebraic data types. By combining them in a point free style, you can build some other useful combinators.

An Aside: Relations, Linear Algebra, Databases

The composition operation described above is not so unfamiliar as it may first appear.

Relation algebra has a great similarity to linear algebra. This connection can be made more clear by considering sparsity patterns of matrices and tensors. Sparsity patterns are a useful abstraction of linear algebraic operations. Instead of considering matrices of numbers, instead the entries are “zero” and “possibly nonzero” or, if you prefer, a matrix of boolean values corresponding to those same questions.

The ordinary row times column matrix multiplication corresponds to relation composition. Replace * with AND and + with OR. If any of the numbers is zero, then multiplying them will result in zero. In summing two numbers, if either is possibly nonzero, then the result is possibly nonzero.

Another interesting way of looking at it is that we are replacing the summation binding form $\sum_i$ with the logical quantifier $\exists_i$. Both introduce a scoped “dummy variable” i and have a lot of syntactic similarity. Other related forms include $\lambda i$, $\forall i$, $\int di$, $\max_i$ .

There is also an analog of the point free relation algebra in linear algebra. Linear algebra has the most widely used point free notation in the world, matrix notation. Consider the expressions $Ax=b$ and $X = ABC$ as compared to $\sum_j A_{ij} x_j = b_i$ and $X_{il} = \sum_{jk} A_{ij} B_{jk} C_{kl}$. Matrix notation is SO much better for certain calculations. Other pieces of the matrix notation include transpose, inverse, Kronecker product, the Khatri-Rao product, and Hadamard product. Their properties are more clear in the index free form in my opinion. I believe even massive tensor expressions can be written index free using these operators. There are also analogies to be drawn between the graphical notations in these different fields.

Databases can be thought of very similarly to sparse matrices. In principle, you could enumerate all the possible values for a column of a database. So you could think of a database as a giant matrix with a 1 if the item is in the database and 0 if not. Databases are very very sparse from this perspective, and you would never store them this way. The join operation is a relative of relational composition, however join usually operates via looking at the column names, whereas our join is position based.

Query optimization in databases has interesting analogs in sparse linear algebra. For example, the Taco compiler http://tensor-compiler.org/ is doing something very akin to a query optimizer.

Inverting Relations

Unlike functions, Relations are always “invertible”. We call this the converse of a relation. When a function is invertible, it corresponds to the converse. In terms of the tuples underlying our representation, it just swaps them. Relations also possess operations trans and untrans that may be thought of as a kind of currying or as a partial inverse on a single parameter.

Orderings can also be lifted to relations $(\leq) = \{(a,b) | a \leq b\}$. The composition of relations also respects the usual composition of ordering.

Nondeterministic choice is sometimes represented in Haskell using Set returning functions a -> [b]. You may recall this from the context of the List monad. In fact in this case, we have an isomorphism as evidenced by tabulateSearch and searchRel.

Similarly partial functions can be reflected into relations

A useful trick is to lift sets/subsets to relations as a diagonal relation. $\{(a,a) | a \in S \}$. Projection onto the set can be achieve by composing with this relation. The identity results if you are talking about the entire set S.

Comparing Relations

We can compare sets by asking if one is a subset of the other $A\subseteq B$ . Relations can also be compared by this operation, which we call relational inclusion.

A subservient notion to this is relational equality.

Relational algebra is chockful of inequality style reasoning, which is richer and slightly more complicated than equality style reasoning. This is one of the benefits of moving from functional descriptions to a relational description.

Relations also form a lattice with respect to these comparisons. What the hell are lattices? In the context of finite relations, lattices may be over powered mathematical machinery, but it really is useful down the line. They give you binary operators that play nice with some kind of ordering, in our case relational inclusion. These two operations are the meet and the join, which find the greatest lower bound and least upper bound of the operands respectively. For our relations, these correspond to the more familiar notion of set intersection and union. The intersection of two sets is the biggest set that is in both of them. The union is the smallest set for which both sets are a subset of it.

Using meet/join vs intersection/union becomes more interesting when the domain is fancier than relations over finite domains. Issues of infinity can make this interesting, or when using a representation that can’t explicitly represent arbitrary unions or intersections, but that instead has to approximate them. My favorite example is polyhedra. Polyhedra are not closed under unions. So in this case the join and union do not coincide. You need to take a convex hull of the union instead, which is the best approximation. Concretely, polyhedra can be represented as a list of their vertices, which generate the polyhedron. There is no way to express a union in this representation. Concatenating the lists represents taking the convex hull of the union.

An additional property that a lattice may possess is a largest and small element, called top ($\top$ ) and bottom ($\bot$). Our finite domain relations do have these.

Relational Division

And now finally we get to one of the most interesting, powerful, and confusing operations: relational division. Relational division is a kind of pseudo inverse to relational composition. In linear algebra, the pseudo inverse is a matrix that does the best job it can to invert another matrix in a least squares sense. If the matrix is actually invertible, it equals the inverse. Relational division does the best job it can to invert a relational composition. Instead of taking least squares as a criteria, it ensures that the result doesn’t over approximate. If you had the inequality $X \cdot Y \subseteq Z$ and you want to solve for X, relational division is the thing that does that. The right division $Q = Z/Y$ is the largest relation such that $Q \cdot Y \subseteq Z$.

A helpful example is the similar operation of division in database tables.

And here is an implementation that I think is correct. I’ve goofed it up a couple times, it is a rather confusing construct.

There also exists a very similar operation of ldiv.

Relational division encapsulates many notions of searching or optimizing. I invite you to read more about it in J.N. Oliveira’s text or the Bird & de Moor text.

Properties and QuickCheck

Relation algebra is so chock full of properties. This is a perfect opportunity for some QuickCheck , a randomized property testing framework. There are so many more to test. I need to dig through to collect up all the identities.

Bits and Bobbles

• Relations over continuous spaces. Vector subspaces (Linear Relations), Polyhedra (Linear inequality relations).
• Non Bool-valued Relations. Replace $\exists_x$ with $\max_x$. The weighted edgelist of a graph is a natural relation. By using composition we can ask about paths. We still have a comparison operator $\subseteq$ which now respects the ordering of weights
• Galois connections are cool.
• Relations combined with recursion schemes. Recursion schemes are the point free way of describing recursion.
• Moving into infinite spaces. How do we cope?
• Faster search. Some relations are best specified by functions, Maps, others, mixes and matching.
• If you “singletonize” relations a la the Agda project https://github.com/scmu/aopa, you get very interesting interconnections with profunctors, which people say are a categorical generalization of relations.
• Point-free DSLs are interesting and pleasant. Many worries about alpha renaming are gone, at the expense of point-free pain. A DSL like this may be necessary to choose good relational query plans

Edit: A follow up post on that type level angle here http://www.philipzucker.com/relational-algebra-with-fancy-types/

References

Edit : A math exchange question about a -> [b] relational type. https://math.stackexchange.com/questions/3360026/can-division-be-expressed-intensionally-in-relation-algebra/3361351#3361351

Edit: An interesting comment and related library from /u/stevana

Why I (as of June 22 2019) think Haskell is the best general purpose language (as of June 22 2019)

Me and my close friends have been interested in starting a project together and I suggested we use Haskell. I do not think the suggestion was received well or perhaps in seriousness. I have a tendency to joke about almost everything and have put forward that we use many interesting but not practical languages in the same tone that I suggest Haskell. This was a tactical mistake. I find myself in despair at the idea I can’t convince my personal friends, who are curious and intellectual people, to use Haskell on a fresh start web project we have complete control over. What hope do I have in the world at large? This brain dump post is meant for them. My suggestion to use Haskell is not just me being an asshole, although that does make it more fun for me. I will now try to explain in all seriousness and in all the honesty that I can muster what my opinions on languages are and why I have them.

Pragmatically can you start a new project using a language you don’t know? This is a problem. A project always has some intrinsic difficulty. Not all projects will survive an extra layer of unnecessary speedbump. But when and how are you supposed to learn new languages? Never is one answer. I disagree with this answer. In this case we have the leg up that I do know Haskell. Perhaps this is a downside in that it will be extra frustrating? It is also easy for me to ask, as using Haskell is not a burden for me, I have already sunk the cost, but a massive learning burden for others.

Is Haskell actually practical for a web application? Short answer: yes. Expect pain though. If your web application is so simple you could rip it out in 100 lines of python, this is such a simple project that it is a good opportunity to learn something new. If it will become large and complex, then I believe Haskell does shine, keeping complexity under control. I base this upon the professional experience making a web application using a Haskell-Purescript stack. For honesty, it wasn’t all good. I recall ripping my hair out threading shit through monad stacks to where it need to go. Yet on the whole, it kept the project sane. I also believe this based on the word of mouth that I believe but could be just cultish ramblings.

I believe that truly dominating properties of Haskell appear in large complex projects. This is difficult to prove in any other way except empirically and the experiments will be so wildly uncontrolled in terms of the project and people involved that no conclusions can truly be drawn. And yet I have faith, and think that personal experience validates this opinion to myself at least. We have to live this life even though truth does not exist. Choices and opinions must be made.

For programs that are going to be a single manageable file and written in one night, it doesn’t matter much what you use in terms of being choked on your own code. At this scale, I still think Haskell is enjoyable and interesting though. Haskell was my doorway into the world of computer science as I now understand it. I hope there are more doorways.

Things about me that may be different from you. Decide for yourself if these aspects of me make our opinions fundamentally incompatible.

• I do have a talent and a like for practically oriented mathematical topics (computational methods, linear algebra, formal methods, calculus, geometric algebra, projective geometry, optimization, etc.). I actually have very little taste at all for mathematical topics that I see no purpose to.
• I do have some desire and taste for esoterica for its own purpose. I cannot exactly characterize what makes some topics acceptable to me and others not.
• A hard learning curve is not necessarily a downside for me. I enjoy the challenge if it is overcomable and worth it on the other side.
• I like weird and different. That is a positive for me, but a negative for many. I might just be a millennial hipster idiot.
• I would LOVE to find a language I think is better than Haskell. I would LOVE to abandon Haskell. Perhaps this already makes me odd. Perhaps I think many people don’t consider the differences between languages to be worth making the switch and the wasted knowledge. The people with this opinion may or may not have tried enough languages.
• I have “drank the koolaid”. I do read what comes out of the Haskell and functional programming community and have a tendency to believe things without strong empirical backing.
• I have been more deeply entwined with Haskell than any other language. Perhaps if I had reached a level of familiarity in another language I’d be as fervent about that one? I don’t believe this is the case.
• While I desire performance, I consciously curb this desire. I am strongly in favor of cleanly written, clear, principled code. This is of course a weighted judgement. I will probably use a 100x performance gain for a 2x decrease in clarity or reusability. This is a result of the problem domains and scale that have interested me in the past and that I anticipate in the future. I STRONGLY believe the world at large over values performance at the expense of other good qualities of code. Or at least over optimizes early.

A Biased List of Pros and Cons

• The feature set is huge and difficult to understand
• Extreme amounts of legacy features that will be even recommended if you read legacy documentation. Kitchen sink.
• The language appears to be almost entirely built out of footguns.
• The syntax is too verbose.
• I have a distaste for mutation.
• I have a distaste for object oriented programming

What do I find good about C++

• It is commonly used. Large community.
• It is possible for it to be very fast
• Kitchen Sink. You can find almost any feature you want here.
• The high level goals of the mind-leaders sound good.
• Very interesting projects are written in C++. HPC things. Scientific computation.
• Template metaprogramming seems very powerful, but arcane.

• The syntax puts me off as being incredibly verbose
• Extreme object oriented focus
• Corporate/Enterprise feel. I am an iconoclast and see myself as a reasonable but slightly rebellious character. Java in my mind brings images of cubicles and white walls. Perhaps this is not fair.

What do I find good about Java

• ?

I’m joking. Sorry Java. But I also kind of mean it. Yes, there are positive aspects to Java.

• Very commonly used and understood. Perhaps the lingua franca
• Incredible library ecosystem
• Numpy and scipy in particular are marvels of the modern age.
• Syntax is basically imperative pseudo-code
• I am personally very familiar with it
• python is the easiest to use language I know.

• Python has no natural tendency for correctness due to the free wheeling dynamically typed character. This is patched up with testing, opt-in type systems.
• I don’t know how to grow as a pythonista. The skill curve flattens out. For some this may be a positive.
• The main way of building new data types is the class system. I think this is ungainly, overly verbose, and not always a good conceptual fit.
• Despite being among the most succinct of common imperative languages, I still think it ends up being too verbose.
• It is slow. This is a negative, although not high on my priorities.

• Very difficult learning curve. Let’s get real. Haskell is a very confusing programming language to get started in for common programming tasks.
• functional programming is weirder than imperative programming.
• The monad paradigm, even once learned is ungainly. Tracking multiple effects is a pain that does not exist in most languages.
• The pain is up front. It is easy to get a sketch of what you want ripped out in python faster than in Haskell (for me). If you want a web server, command line tool, optimization problem, curve fitter, I can rip all of these out faster in python than I can in Haskell. As a psychological thing, this feels awful. For a small scale project, unless toying with Haskell itself or one of its domain expertises like implementing DSLs, python is the easier and correct choice. Python is a scripting language. I’d make the switch at two screens worth of code.
• I think laziness is confusing and easy to shoot yourself with.
• Haskell is not the fastest language, although faster than python.
• Concern for there not being jobs and interestingly on the converse side, no people to hire. There is a catch-22 there. There is a set of people that would KILL for a Haskell job.
• There are vocal smug assholes who use Haskell and push it. I am smug, I hope only mildly an asshole.

• The number one reason is that there is something ephemeral that I just like. I am not naturally inclined to analyze such things. When I like a movie or don’t like it, it just happens, and if forced to explain why, I’ll bullshit.
• Errors and bugs are to a shocking degree caught at compile time. More than any other language I have experience with does Haskell code run correctly without hidden bugs if it compiles. I am not claiming this is absolute but it is all the more incredible the degree to which it is true since it didn’t literally have to be this way. I have done major rewiring of a data structure in a project. The compiler guided my hand to exactly the positions that needed to be adjusted. Could C++ do this? Yes, to some degree. I believe that the tendencies of C++ programming make it less satisfactory on this point.
• Types are an incredible design tool. I find designing the types of my program to be an extremely enjoyable and helpful activity. Much more so than box and wire class and interface diagrams. A good function name and a type signature basically entirely constrains the behavior of a function. Types can be quickly and completely be given to the compiler and machine enforced.
• The pain that Monads cause are pains you should be feeling. They are the pain of explicitness which I 70% choose over the pain of not knowing what the fuck a function might do, and not enabling the compiler to enforce that. If something is capable of mutating state, it should say so in goddamn huge purple letters.
• Haskell is more than fast enough. It isn’t that even people don’t care. The Haskell community at large cares a lot more for performance than I do, and I reap the dividends. The people in charge of the compiler and the main libraries are goddamn wizards who’s work I get to benefit from. This is true of all languages perhaps.
• Laziness is very cool. At the beginning I thought it was INCREDIBLY awesome and inconceivable that I could manipulate an infinite list.
• The way of specifying new data types is so succinct and so incredible. Pattern matching is SO GODDAMN good for some tasks.
• Haskell has a paradigm of small combinators. It is common to build a sequence of very small natural functions for the domain and then build larger and larger things out of them. This is good for reusability and conceptual clearness.
• Extreme preference for Immutability. As part of keeping what you must keep in your head or remember small while programming, immutability is an insane win. You think you know what you want now. You know you could just tweak this variable here, make a special variable over here. You can reason about how to make this all work and be correct now. You will not in a month. Your coworkers will mess it all up too.
• Haskell code is generic by default. This allows the same code to be reused in many situations
• The standard typeclass hierarchy is extremely well thought out and powerful. To some degree it is unnecessary in other languages. The difference between Functor, Applicative, Monad, and Traversable makes little sense in languages with unconstrained mutations and effects.
• Haskell paradigms are inspired by mathematics, and I have great faith in mathematics. The concepts behind Haskell feel closer to discoveries rather than inventions. Imperative programming speaks in a language formed for the accidental nature of the machines we have. Functional programming is a language closer to mathematics, which I believe is closer to how the human mind works, and closer to what the problem at hand actually is.
• Complexity scales. It is my belief, perhaps unverified, that as a project grows larger, the insane miserable churn and mental overhead grows slower in a Haskell project than in other languages. I do not have strong empirical evidence to this assertion. Word of mouth (of Haskellers).
• The ceiling on Haskell is extremely high. You can continue to learn and get better, gaining more and more power. I do not currently see the end of this.
• When I do reach for some new library, I am very often impressed by how thoughtfully built it is. Haskell itself is INCREDIBLY thoughtfully built.
• The haskell community is very excited and they have many things to teach. There are many Haskellers out there who are very welcoming, kind, and intelligent.
• Haskell does have some cache. I am not immune to wanting to seem smart. If the world thought that only idiots use Haskell, that would offput me some. That the world things that only impractical ivory tower smug weenies use Haskell does offput me, although I perhaps embrace it belligerently.
• The Haskell library ecosystem is strong. Less strong than python, but much better than some of the other languages that intrigue my eye. There is functionality somewhere for most common tasks.
• Haskell is used industrially. I live in an echo chamber, but by and large the industrial users of Haskell sing its praises.

For context, this is my programming journey:

My history may not be as convincing as someone who spent 20 years as a professional C++ dev and then switched, but I have at least experienced different paradigms and languages and found Haskell the most to my liking.

Random Thoughts

I am now trying to push myself into comfort in Julia, Rust, Agda, Coq, OCaml, all languages I feel show promise for different reasons. To my knowledge Haskell is a better choice than these as a general purpose tool for pragmatic reasons. Haskell’s library ecosystem is strong and performance is good. These are points against agda and coq. Julia has a focus on scientific programming.

Rust might be a good compromise. I consider it a trojan horse for useful programming language features that I associate with functional languages. It claims to performance and being an acceptable systems level language, which appeals to some. The syntax does not scare anyone off. The library ecosystem seems good and the community strong. I did find myself missing Haskell features in Rust though, am personally much less familiar with it. I think the design of Rust weights more to performance than is warranted in most applications and has less descriptive and abstraction power than Haskell, qualities that I prioritize. This opinion is not strongly held.

What makes Haskells types any better or worse than C? At the beginning many of the features of Haskell seem like magic. But as time has worn on, I can see how the features can be emulated with just some slight syntactic and conceptual overhead in other languages. This slight overhead is enough though. A language is more than just it’s syntax. It is also its idioms. It is also they way it makes people think. Language is almost a prerequisite for thought. You cannot even conceive of the ways there are to express yourself before learning.

What exactly makes a language “good”? This is a question poorly phrased enough to have no answer. Excel can be an excellent language for many tasks. Easy to learn. Very powerful. Yet, it is not considered a good general purpose programming language. Library ecosystem is extremely important. Specialized languages are often the best choice for special problem domains, at the expense of learning them, or eventually finding incompatibility of what you want from what they designed for.

What makes abstractions “good”. Why do I have queasiness about object oriented-programmming. Java, I think basically. I, overeagerly have gone down the road of trying to design deep subclass hierarchies, which is not OO at it’s best. Zebra is a Quadruped is an Animal is Alive kind of stuff. I believe object oriented in an interesting principle. I hear about SmallTalk and Common Lisp doing object oriented right and I am duly intrigued. There has been some recent work in Haskell about how to do objects in a way aesthetically compatible with Haskell. I think object oriented has been over used and abused. I think it is a poor conceptual fit for many situations. I think it tends to make non reusable code. I think the form that it takes in C++ and Java development is arcane horseshit.

I deserve almost no opinion about Java or C++, having not done sufficient that much in them. Yet, I must state my opinions, take them as you will, for I do in fact hold them strongly. I have worked on a network simulator and a robotics framework in C++, but begrudgingly. I have done a very small amount of Java development for a personal project and some Processing sketches. My coworker was a 10 year professional Java dev before switching to Scala then Haskell. He despises Java now. Highly anecdotal, and he is a similar iconoclastic character like me. Nevertheless, this also informs my opinion. I have been reading Bjarne Stroustrup’s book (his stated goals and what he claims C++ achieves are admirable and one can’t argue he hasn’t changed the world) and actually find C++ rather interesting, especially in the sense that many projects that I find interesting are written in C++, I just don’t want to myself work in the language.

https://news.ycombinator.com/item?id=17114308 Ah Hacker News. Always a sunny worldview.

Hacker news discussion of this post:

https://news.ycombinator.com/item?id=20258218#20271511

Lens as a Divisibility Relation: Goofin’ Off With the Algebra of Types

Types have an algebra very analogous to the algebra of ordinary numbers (video). This is the basic table of correspondences. Code with all the extensions available here.

One way to see that this makes sense is by counting the cardinality of types built out of these combinators. Unit is the type with 1 inhabitant. Void has 0 inhabitants. If a has $n$ and b has $m$ possible values, then Either a b has $n + m$ inhabitants, (a,b) has $n*m$ and there are $n^m$ possible tabulations of a->b. We’re gonna stick to just polynomials for the rest of this, ignoring a->b.

Another way of looking at this is if two finitely inhabited types have the same number of inhabitants, then the types can be put into an isomorphism with each other. In other words, types modulo isomorphisms can be thought as representing the natural numbers. Because of this, we can build a curious proof system for the natural numbers using ordinary type manipulation.

In addition, we also get a natural way of expressing and manipulating polynomials.Polymorphic types can be seen as being very similar to polynomial expressions with natural coefficients N[x]. The polymorphic type variables have the ability to be instantiated to any value, corresponding to evaluating the polynomial for some number.

The Lens ecosystem gives some interesting combinators for manipulating this algebra. The type Iso' a b contains isomorphisms. Since we’re only considering types up to isomorphism, this Iso' represents equality. We can give identity isomorphisms, compose isomorphisms and reverse isomorphisms.

We can already form a very simple proof.

Now we’ll add some more combinators, basically the axioms that the types mod isos are a commutative semiring. Semirings have an addition and multiplication operator that distribute over each other. It is interesting to note that I believe all of these Iso' actually are guaranteed to be isomorphisms ( to . from = id and from . to = id ) because of parametricity. from and to are unique ignoring any issues with bottoms because the polymorphic type signature is so constraining. This is not usually guaranteed to be true in Haskell just from saying it is an Iso'. If I give you an Iso' Bool Bool it might actually be the iso (const True) (const True) for example, which is not an isomorphism.

There are also combinators for lifting isomorphisms into bifunctors: firsting, seconding, and bimapping. These are important for indexing into subexpressions of our types in a point-free style.

Here is a slightly more complicated proof now available to us.

We can attempt a more interesting and difficult proof. I developed this iteratively using . _ hole expressions so that GHC would tell me what I had manipulated my type to at that point in my proof.

The proof here is actually pretty trivial and can be completely automated away. We’ll get to that later.

If Iso' is equality, what about the other members of the Lens family? Justin Le says that Lens s a are witness to the isomorphism of a type s to the tuple of something and a. Prism witness a similar thing for sums. Once we are only considering types mod isos, if you think about it, these are expressions of two familiar relations on the natural numbers: the inequality relation and the divisibility relation

Mathematically, these relations can be composed with equalities, just like in the lens hierarchy Lens and Prism can be composed with Iso. Both form a category, since they both have id and (.).

Here are a couple identities that we can’t derive from these basic combinators. There are probably others. Woah-ho, my bad. These are totally derivable using id_mul, id_plus, mul_zero, _1, _2, _Left, _Right.

Pretty neat! Random thoughts and questions before we get into the slog of automation:

• Traversal is the “is polynomial in” relation, which seems a rather weak statement on it’s own.
• Implementing automatic polynomial division is totally possible and interesting
• What is the deal with infinite types like [a]? Fix. I suppose this is a theory of formal power series / combinatorial species. Fun combinatorics, generatingfunctionology. Brent Yorgey did his dissertation on this stuff. Wow. I’ve never really read this. It is way more relevant than I realized.
• Multivariate polynomial algorithms would also be interesting to explore (Grobner basis, multivariate division)
• Derivatives of types and zippers – Conor McBride
• Negative Numbers and Fractions?
• Lifting to rank-1 types. Define Negative and Fractions via Galois connection?

Edit: /u/carette (wonder who that is 😉 ) writes:

“You should dig into
J Carette, A Sabry Computing with semirings and weak rig groupoids, in Proceedings of ESOP 2016, p. 123-148. Agda code in https://github.com/JacquesCarette/pi-dual/tree/master/Univalence. A lot of the algebra you develop is there too.

If you hunt around in my repos, you’ll also find things about lenses, exploring some of the same things you mention here.”

Similar ideas taken further and with more sophistication. Very interesting. Check it out.

Automation

Our factor example above was quite painful, yet the theorem was exceedingly obvious by expansion of the left and right sides. Can we automate that? Yes we can!

Here’s the battle plan:

• Distribute out all expressions like $a*(b+c)$ so that all multiplication nodes appear at the bottom of the tree.
• Reduce the expression by absorbing all stupid $a*1$, $a*0$, $a+0$ terms.
• Reassociate everything to the right, giving a list like format
• Sort the multiplicative terms by power of the variable

Once we have these operations, we’ll combine them into a canonical operation. From there, most equality proofs can be performed using the rewrite operation, which merely puts both sides into canonical form

Once we have those, the painful theorem above and harder ones becomes trivial.

Now we’ll build the Typeclasses necessary to achieve each of these aims in order. The Typeclass system is perfect for what we want to do, as it builds terms by inspecting types. It isn’t perfect in the sense that typeclass pattern matching needs to be tricked into doing what we need. I have traded in cleverness and elegance with verbosity.

In order to make our lives easier, we’ll need to tag every variable name with a newtype wrapper. Otherwise we won’t know when we’ve hit a leaf node that is a variable. I’ve used this trick before here in an early version of my faking Compiling to Categories series. These wrappers are easily automatically stripped.

A common pattern I exploit is to use a type family to drive complicated recursion. Closed type families allow more overlap and default patterns which is very useful for programming. However, type families do not carry values, so we need to flip flop between the typeclass and type family system to achieve our ends.

Here is the implementation of the distributor Dist. We make RDist and LDist typeclasses that make a sweep of the entire tree, using ldist and rdist as makes sense. It was convenient to separate these into two classes for my mental sanity. I am not convinced even now that I have every case. Then the master control class Dist runs these classes until any node that has a (*) in it has no nodes with (+) underneath, as checked by the HasPlus type family.

Next is the Absorb type class. It is arranged somewhat similarly to the above. Greedily absorb, and keep doing it until no absorptions are left. I think that works.

The Associators are a little simpler. You basically just look for the wrong association pattern and call plus_assoc or mul_assoc until they don’t occur anymore, then recurse. We can be assured we’re always making progress if we either switch some association structure or recurse into subparts.