I am happy to announce the release of OASIS v0.4.7.

OASIS is a tool to help OCaml developers to integrate configure, build and install systems in their projects. It should help to create standard entry points in the source code build system, allowing external tools to analyse projects easily.

This tool is freely inspired by Cabal which is the same kind of tool for Haskell.

You can find the new release here and the changelog here. More information about OASIS in general on the OASIS website.

Pull request for inclusion in OPAM is pending.

Here is a quick summary of the important changes:

- Drop support for OASISFormat 0.2 and 0.1.
- New plugin "omake" to support build, doc and install actions.
- Improve automatic tests (Travis CI and AppVeyor)
- Trim down the dependencies (removed ocaml-gettext, camlp4, ocaml-data-notation)

Features:

- findlib_directory (beta): to install libraries in sub-directories of findlib.
- findlib_extra_files (beta): to install extra files with ocamlfind.
- source_patterns (alpha): to provide module to source file mapping.

This version contains a lot of changes and is the achievement of a huge amount of work. The addition of OMake as a plugin is a huge progress. The overall work has been targeted at making OASIS more library like. This is still a work in progress but we made some clear improvement by getting rid of various side effect (like the requirement of using "chdir" to handle the "-C", which leads to propage ~ctxt everywhere and design OASISFileSystem).

I would like to thanks again the contributor for this release: Spiros Eliopoulos, Paul Snively, Jeremie Dimino, Christopher Zimmermann, Christophe Troestler, Max Mouratov, Jacques-Pascal Deplaix, Geoff Shannon, Simon Cruanes, Vladimir Brankov, Gabriel Radanne, Evgenii Lepikhin, Petter Urkedal, Gerd Stolpmann and Anton Bachin.

**Overview**

Capital Match is a leading marketplace lending and invoice financing platform in Singapore. Our in-house platform, mostly developed in Haskell, has in the last year seen more than USD 10 million business loans processed with a strong monthly growth (current rate of USD 1.5-2.5 million monthly). We are also eyeing expansion into new geographies and product categories. Very exciting times!

We have just secured another funding round to build a world-class technology as the key business differentiator. The key components include credit risk engine, seamless banking integration and end-to-end product automation from loan origination to debt collection.

**Responsibilities**

We are looking to hire a software engineer with a minimum of 2-3 years coding experience. The current tech team includes a product manager and 3 software engineers. We are currently also in the process of hiring CTO.

The candidate should have been involved in a development of multiple web-based products from scratch. He should be interested in all aspects of the creation, growth and operations of a secure web-based platform: front-to-back features development, distributed deployment and automation in the cloud, build and test automation etc.

Background in fintech and especially lending / invoice financing space would be a great advantage.

**Requirements**

Our platform is primarily developed in Haskell with an Om/ClojureScript frontend. We are expecting our candidate to have experience working with a functional programming language e.g. Haskell/Scala/OCaml/F#/Clojure/Lisp/Erlang.

Deployment and production is managed with Docker containers using standard cloud infrastructure so familiarity with Linux systems, command-line environment and cloud-based deployment is mandatory. Minimum exposure to and understanding of XP practices (TDD, CI, Emergent Design, Refactoring, Peer review and programming, Continuous improvement) is expected.

We are looking for candidates that are living in or are willing to relocate to Singapore.

**Offer**

We offer a combination of salary and equity depending on experience and skills of the candidate.

Most expats who relocate to Singapore do not have to pay their home country taxes and the local tax rate in Singapore is more or less 5% (effective on the proposed salary range).

Visa sponsorship will be provided.

Singapore is a great place to live, a vibrant city rich with diverse cultures, a very strong financial sector and a central location in Southeast Asia.

Get information on how to apply for this position.

**About SAP**

SAP is a market leader in enterprise application software, helping companies of all sizes and industries run better. SAP empowers people and organizations to work together more efficiently and use business insight more effectively. SAP applications and services enable our customers to operate profitably, adapt continuously, and grow sustainably.

**What you'll do:**

You will be a member of the newly formed Scala development experience team. You will support us with the design and development of a Scala and cloud-based business application development and runtime platform. The goal of the platform is to make cloud-based business application development in the context S/4 HANA as straight-forward as possible. The team will be distributed over Berlin, Potsdam, Walldorf and Bangalore.

Your tasks as a (Senior) Scala Developer will include:

- Design and development of libraries and tools for business application development
- Design and development of tools for operating business applications
- Explore, understand, and implement most recent technologies
- Contribute to open source software (in particular within the Scala ecosystem)

**Required skills:**

- Master’s degree in computer science, mathematics, or related field
- Excellent programming skills and a solid foundation in computer science with strong competencies in data structures, algorithms, databases, and software design
- Solid understanding of object-oriented concepts and basic understanding functional programming concepts
- Good knowledge in Scala, Java, C++, or similar object-oriented programming languages
- Strong analytical skills
- Reliable and open-minded with strong team working skills, determined to reach a goal in time as well as the ability to work independently and to prioritize
- Ability to get quickly up-to-speed in a complex, new environment
- Proficiency in spoken and written English

**Beneficial skills**

- Ph.D. in computer science
- Solid understanding of functional programming concepts
- Good knowledge in Scala, OCaml, SML, or Haskell
- Experience with Scala and Scala.js
- Experience with meta programming in Scala, e.g., using Scala’s macro system
- Knowledge on SAP technologies and products
- Experiences with the design of distributed systems, e.g., using Akka

**What we offer**

- Modern and innovative office locations
- Free lunch and free coffee
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- Fitness room with a climbing wall
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- Friendly colleagues and the opportunity to work within a highly diverse team which has expert knowledge in a wide range of technologies

Get information on how to apply for this position.

Mike Shulman just wrote a very nice blog post on what is a formal proof. I much agree with what he says, but I would like to offer my own perspective. I started writing it as a comment to Mike’s post and then realized that it is too long, and that I would like to have … Continue reading What is a formal proof?

Mike Shulman just wrote a very nice blog post on what is a formal proof. I much agree with what he says, but I would like to offer my own perspective. I started writing it as a comment to Mike’s post and then realized that it is too long, and that I would like to have it recorded independently as well. Please read Mike’s blog post first.

Just as Mike, I am discussing here formal proofs from the point of view of proof assistants, i.e., what criteria need to be satisfied by the things we call “formal proofs” for them to serve their intended purpose, which is: to convince machines (and indirectly humans) of mathematical truths. Just as Mike, I shall call a (formal) proof a *complete* derivation tree in a formal system, such as type theory or first-order logic.

What Mikes calls an *argument* I would prefer to call a *proof representation*. This can be any kind of concrete representation of the actual formal proof. The representation may be very indirect and might require a lot of effort to reconstruct the original proof. Unless we deal with an extremely simple formal system, there is always the possibility to have *invalid representations*, i.e., data of the correct datatype which however does not represent a proof.

I am guaranteed to reinvent the wheel here, at least partially, since many people before me thought of the problem, but here I go anyway. Here are (some) criteria that formal proofs should satisfy:

**Reproducibility:**it should be possible to replicate and communicate proofs. If I have a proof it ought to be possible for me to send you a copy of the proof.**Objectivity:**all copies of the same proof should represent the same piece of information, and there should be no question what is being represented.**Verifiability:**it should be possible to recognize the fact that something is a proof.

There is another plausible requirement:

**Falsifiability:**it should be possible to recognize the fact that something is*not*a proof.

Unlike the other three requirements, I find falsifiability questionable. I have received too many messages from amateur mathematicians who could not be convinced that their proofs were wrong. Also, mathematics is a cooperative activity in which mistakes (both honest and dishonest) are easily dealt with – once we expand the resources allocated to verifying a proof we simply give up. An adversarial situation, such as proof carrying code, is a different story with a different set of requirements.

The requirements impose conditions on how formal proofs in a proof assistant might be designed. Reproducibility dictates that proofs should be easily accessible and communicable. That is, they should be pieces of digital information that are commonly handled by computers. They should not be prohibitively large, of if they are, they need to be suitably compressed, lest storage and communication become unfeasible. Objectivity is almost automatic in the era of crisp digital data. We will worry about Planck-scale proof objects later. Verifiability can be ensured by developing and implementing algorithms that recognize correct representations of proofs.

This post grew out of a comment that I wanted to make about a particular statement in Mike’s post. He says:

“… for a proof assistant to honestly call itself an

implementationof that formal system, it ought to include, somewhere in its internals, some data structure that represents those proofs reasonably faithfully.”

This requirement is too stringent. I think Mike is shooting for some combination of reproducibility and verifiability, but explicit storage of proofs in raw form is only one way to achieve them. What we need instead is *efficient communication* and *verification *of (communicated) proofs. These can both be achieved without storage of proofs in explicit form.

Proofs may be stored and communicated in implicit form, and proof assistants such as Coq and Agda do this. Do not be fooled into thinking that Coq gives you the “proof terms”, or that Agda aficionados type down actual complete proofs. Those are not the derivation trees, because they are missing large subtrees of equality reasoning. Complete proofs are too big to be communicated or stored in memory (or some day they will be), and little or nothing is gained by storing them or re-verifying their complete forms. Instead, it is better to devise compact representations of proofs which get *elaborated* or *evaluated* into actual proofs on the fly. Mike comments on this and explains that Coq and Agda both involve a large amount of elaboration, but let me point out that even the elaborated stuff is still only a shadow of the actual derivation tree. The data that gets stored in the Coq .vo file is really a bunch of instructions for the proof checker to easily reconstruct the proof using a specific algorithm. The *actual* derivation tree is implicit in the execution trace of the proof checker, stored in the space-time continuum and inaccessible with pre-Star Trek technology. It does not matter that we cannot get to it, because the whole process is replicable. If we feel like going through the derivation tree again, we can just run the proof assistant again.

I am aware of the fact that people strongly advocate some points which I am arguing against, two of which might be:

- Proofs assistants must provide proofs that can be independently checked.
- Proof checking must be
*decidable*, not just*semi-decidable.*

As far as I can tell, nobody actually subscribes to these in practice. (Now that the angry Haskell mob has subsided, I feel like I can take a hit from an angry proof assistant mob, which the following three paragraphs are intended to attract. What I *really* want the angry mob to think about deeply is how their professed beliefs match up with their practice.)

First, nobody downloads compiled .vo files that contain the proof certificates, we all download other people’s original .v files and compile them ourselves. So the .vo files and proof certificates are a double illusion: they do not contain actual proofs but half-digested stuff that may still require a lot of work to verify, and nobody uses them to communicate or verify proofs anyhow. They are just an optimization technique for faster loading of libraries. The *real* representations of proofs are in the .v files, and those can only be *semi-*checked for correctness.

Second, in practice it is irrelevant whether checking a proof is decidable because the elaboration phase and the various proof search techniques are possibly non-terminating anyhow. If there are a couple of possibly non-terminating layers on top of the trusted kernel, we might as well let the kernel be possibly non-terminating, too, and instead squeeze some extra expressivity and efficiency from it.

Third, and still staying with decidability of proof checking, what actually *is* annoying are uncontrollable or unidentifiable sources of inefficiency. Have you ever danced a little dance around Coq or Agda to cajole its *terminating* normalization procedure into finishing before getting run over by Andromeda? Bow to the gods of decidable proof checking.

It is far more important that *cooperating* parties be able to communicate and verify proofs efficiently, than it is to be able to tell whether an *adversary* is wasting our time. Therefore, proofs should be, and in practice are communicated in the most flexible manner possible, as programs. LCF-style proof assistants embrace this idea, while others move slowly towards it by giving the user ever greater control over the internal mechanisms of the proof assistant (for instance, witness Coq’s recent developments such as partial user control over the universe mechanism, or Agda’s rewriting hackery). In an adversarial situations, such as proof carrying code, the design requirements for formalized proofs are completely different from the situation we are considering.

We do not expect humans to memorize every proof of every mathematical statement they ever use, nor do we imagine that knowledge of a mathematical fact is the same thing as the proof of it. Humans actually memorize “proof ideas” which allow them to replicate the proofs whenever they need to. Proof assistants operate in much the same way, for good reasons.

This post is going to draw an angry Haskell mob, but I just have to say it out loud: I have never seen a definition of the so-called category Hask and I do not actually believe there is one until someone does some serious work. Let us look at the matter a bit closer. The Haskell … Continue reading Hask is not a category

This post is going to draw an angry Haskell mob, but I just have to say it out loud: I have never seen a definition of the so-called category Hask and I do not actually believe there is one until someone does some serious work.

Let us look at the matter a bit closer. The Haskell wiki page on Hask says:

The objects of Hask are Haskell types, and the morphisms from objects `A`

to `B`

are Haskell functions of type `A -> B`

. The identity morphism for object `A`

is `id :: A -> A`

, and the composition of morphisms `f`

and `g`

is `f . g = \x -> f (g x)`

.

Presumably “function” here means “closed expression”. It is then immediately noticed that there is a problem because the supposed identity morphisms do not actually work correctly: `seq undefined () = undefined`

and `seq (undefined . id) () = ()`

, therefore we do not have `undefined . id = undefined`

.

The proposed solution is to equate `f :: A -> B`

and `g :: A -> B`

when `f x = g x`

for all `x :: A`

. Again, we may presume that here `x`

ranges over all closed expressions of type `A`

. But this begs the question: *what does f x = g x mean?* Obviously, it cannot mean “syntactically equal expressions”. If we had a notion of observational or contextual equivalence then we could use that, but there is no such thing until somebody provides an operational semantics of Haskell. Written down, in detail, in standard form.

The wiki page gives two references. One is about the denotational semantics of Haskell, which is just a certain category of continuous posets. That is all fine, but such a category is not the syntactic category we are looking for. The other paper is a fine piece of work that uses denotational semantics to prove cool things, but does not speak of any syntactic category for Haskell.

There are several ways in which we could resolve the problem:

- If we define a notion of observational or contextual equivalence for Haskell, then we will know what it means for two expressions to be indistinguishable. We can then use this notion to equate indistinguishable morphisms.
- We could try to define the equality relation more carefully. The wiki page does a first step by specifying that at a function type equality is the extensional equality. Similarly, we could define that two pairs are equal if their components are equal, etc. But there are a lot of type constructors (including recursive types) and you’d have to go through them, and define a notion of equality on all of them. And after that, you need to show that this notion of equality actually gives a category. All the while, there will be a nagging doubt as to what it all means, since there is no operational semantics of Haskell.
- We could import a category-theoretic structure from a denotational semantics. It seems that denotational semantics of Haskell actually exists and is some sort of a category of domains. However, this would just mean we’re restricting attention to a subcategory of the semantic category on the definable objects and morphisms. There is little to no advantage of doing so, and it’s better to just stick with the semantic category.

Until someone actually does some work, **there is no Hask**! I’d delighted to be wrong, but I have not seen a complete construction of such a category yet.

Perhaps you think it is OK to pretend that something is a category when it is not. In that case, you would also pretend that the Haskell monads are actual category-theoretic monads. I recall a story from one of my math professors: when she was still a doctoral student she participated as “math support” in the construction of a small experimental nuclear reactor in Slovenia. One of the physicsts asked her to estimate the value of the harmonic series $1 + 1/2 + 1/3 + \cdots$ to four decimals. When she tried to explain the series diverged, he said “that’s ok, let’s just pretend it converges”.

**Supplemental: ** Of the three solutions mentioned above I like the best the one where we give Haskell an operational semantics. It’s more or less clear how we would do this, after all Haskell is more or less a glorified PCF. However, the thing that worries me is `seq`

. Because of it `undefined`

and `undefined . id`

are *not* observationally equivalent, which means that we cannot use observational equivalence for equality of morphisms. We could try the wiki definition: `f :: A -> B`

and `g :: A -> B`

represent the same morphisms if `f x`

and `g x`

are observationally equivalent for all closed expressions `x :: A`

. But then we need to prove something after that to know that we really have a category. For instance, I do not find it obvious anymore that programs which involve seq behave nicely. And what happens with higher-order functions, where observational equivalence and extensional equality get mixed up, is everything still holding water? There are just too many questions to be answered before we have a category.

**Supplemental II:** Now that the mob is here, I can see certain patterns in the comments, so I will allow myself replying to them en masse by supplementing the post. I hope you all will notice this. Let me be clear that I am not arguing against the usefulness of category-theoretic thinking in programming. In fact, I support programming that is informed by abstraction, as it often leads to new insights and helps gets things done correctly. (And anyone who knows my work should find this completely obvious.)

Nor am I objecting to “fast & loose” mode of thinking while investigating a new idea in Haskell, that is obviously quite useful as well. I am objecting to:

- The fact the the Haskell wiki claims there is such a thing as “the category Hask” and it pretends that everything is ok.
- The fact that some people find it acceptable to defend broken mathematics on the grounds that it is useful. Non-broken mathematics is also useful, as well as correct. Good engineers do not rationalize broken math by saying “life is tough”.

Anyhow, we do not need the Hask category. There already are other categories into which we can map Haskell, and they explain things quite well. It is ok to say “you can think of Haskell as a sort of category, but beware, there are technical details which break this idea, so you need to be a bit careful”. It is not ok to write on the Haskell wiki “Hask is a category”. Which is why I put up this blog post, so when people Google for Hask they’ll hopefully find the truth behind it.

**Supplemental III**: On Twitter people have suggested some references that provide an operational semantics of Haskell:

- John Launchbury: A natural semantics for lazy evaluation
- Alan Jeffrey: A fully abstract semantics for concurrent graph reduction

Can we use these to define a suitable notion of equality of morphisms? (And let’s forget about `seq`

for the time being.)

Software Developer

Jane Street is a proprietary quantitative trading firm, focusing primarily on trading equities and equity derivatives. We use innovative technology, a scientific approach, and a deep understanding of markets to stay successful in our highly competitive field. We operate around the clock and around the globe, employing over 400 people in offices in New York, London and Hong Kong.

The markets in which we trade change rapidly, but our intellectual approach changes faster still. Every day, we have new problems to solve and new theories to test. Our entrepreneurial culture is driven by our talented team of traders and programmers. At Jane Street, we don't come to work wanting to leave. We come to work excited to test new theories, have thought-provoking discussions, and maybe sneak in a game of ping-pong or two. Keeping our culture casual and our employees happy is of paramount importance to us.

We are looking to hire great software developers with an interest in functional programming. OCaml, a statically typed functional programming language with similarities to Haskell, Scheme, Erlang, F# and SML, is our language of choice. We've got the largest team of OCaml developers in any industrial setting, and probably the world's largest OCaml codebase. We use OCaml for running our entire business, supporting everything from research to systems administration to trading systems. If you're interested in seeing how functional programming plays out in the real world, there's no better place.

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You can learn more about Jane Street and our technology from our main site, janestreet.com. You can also look at a a talk given at CMU about why Jane Street uses functional programming (http://ocaml.janestreet.com/?q=node/61), and our programming blog (http://ocaml.janestreet.com).

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