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Native support for differential programming has been added to Swift for the Swift for Tensorflow project. Julia has similar with Zygote.
What exactly is differentiable programming?
what does it enable? Wikipedia says
the programs can be differentiated throughout
but what does that mean?
how would one use it (e.g. a simple example)?
and how does it relate to automatic differentiation (the two seem conflated a lot of the time)?
I like to think about this question in terms of user-facing features (differentiable programming) vs implementation details (automatic differentiation).
From a user's perspective:
"Differentiable programming" is APIs for differentiation. An example is a def gradient(f) higher-order function for computing the gradient of f. These APIs may be first-class language features, or implemented in and provided by libraries.
"Automatic differentiation" is an implementation detail for automatically computing derivative functions. There are many techniques (e.g. source code transformation, operator overloading) and multiple modes (e.g. forward-mode, reverse-mode).
Explained in code:
def f(x):
return x * x * x
∇f = gradient(f)
print(∇f(4)) # 48.0
# Using the `gradient` API:
# ▶ differentiable programming.
# How `gradient` works to compute the gradient of `f`:
# ▶ automatic differentiation.
I never heard the term "differentiable programming" before reading your question, but having used the concepts noted in your references, both from the side of creating code to solve a derivative with Symbolic differentiation and with Automatic differentiation and having written interpreters and compilers, to me this just means that they have made the ability to calculate the numeric value of the derivative of a function easier. I don't know if they made it a First-class citizen, but the new way doesn't require the use of a function/method call; it is done with syntax and the compiler/interpreter hides the translation into calls.
If you look at the Zygote example it clearly shows the use of prime notation
julia> f(10), f'(10)
Most seasoned programmers would guess what I just noted because there was not a research paper explaining it. In other words it is just that obvious.
Another way to think about it is that if you have ever tried to calculate a derivative in a programming language you know how hard it can be at times and then ask yourself why don't they (the language designers and programmers) just add it into the language. In these cases they did.
What surprises me is how long it to took before derivatives became available via syntax instead of calls, but if you have ever worked with scientific code or coded neural networks at at that level then you will understand why this is a concept that is being touted as something of value.
Also I would not view this as another programming paradigm, but I am sure it will be added to the list.
How does it relate to automatic differentiation (the two seem conflated a lot of the time)?
In both cases that you referenced, they use automatic differentiation to calculate the derivative instead of using symbolic differentiation. I do not view differentiable programming and automatic differentiation as being two distinct sets, but instead that differentiable programming has a means of being implemented and the way they chose was to use automatic differentiation, they could have chose symbolic differentiation or some other means.
It seems you are trying to read more into what differential programming is than it really is. It is not a new way of programming, but just a nice feature added for doing derivatives.
Perhaps if they named it differentiable syntax it might have been more clear. The use of the word programming gives it more panache than I think it deserves.
EDIT
After skimming Swift Differentiable Programming Mega-Proposal and trying to compare that with the Julia example using Zygote, I would have to modify the answer into parts that talk about Zygote and then switch gears to talk about Swift. They each took a different path, but the commonality and bottom line is that the languages know something about differentiation which makes the job of coding them easier and hopefully produces less errors.
About the Wikipedia quote that
the programs can be differentiated throughout
At first reading it seems nonsense or at least lacks enough detail to understand it in context which is why I am sure you asked.
In having many years of digging into what others are trying to communicate, one learns that unless the source has been peer reviewed to take it with a grain of salt, and unless it is absolutely necessary to understand, then just ignore it. In this case if you ignore the sentence most of what your reference makes sense. However I take it that you want an answer, so let's try and figure out what it means.
The key word that has me perplexed is throughout, but since you note the statement came from Wikipedia and in Wikipedia they give three references for the statement, a search of the word throughout appears only in one
∂P: A Differentiable Programming System to Bridge Machine Learning and Scientific Computing
Thus, since our ∂P system does not require primitives to handle new
types, this means that almost all functions and types defined
throughout the language are automatically supported by Zygote, and
users can easily accelerate specific functions as they deem necessary.
So my take on this is that by going back to the source, e.g. the paper, you can better understand how that percolated up into Wikipedia, but it seems that the meaning was lost along the way.
In this case if you really want to know the meaning of that statement you should ask on the Wikipedia talk page and ask the author of the statement directly.
Also note that the paper referenced is not peer reviewed, so the statements in there may not have any meaning amongst peers at present. As I said, I would just ignore it and get on with writing wonderful code.
You can guess its definition by application of differentiability.
It's been used for optimization i.e. to calculate minimum value or maximum value
Many of these problems can be solved by finding the appropriate function and then using techniques to find the maximum or the minimum value required.
Are there any common terms for the difference between data validation you can do on, say, an object in and of itself, and validation that requires access to some sort of external resources?
For example, if I have a user record, I can check things like "Is username present?" "Is Username at least n characters long?" etc., without requiring any additional context. But as soon as I want to do something like "Is username available?" It requires checking against other records in my system.
I'm just wondering if there are any good terms for describing the difference in these types of scenarios? "Static analysis" vs. "run-time checking" sort of fits, but it's clearly not correct.
I don't really know of any widely accepted terms for these different kinds of validation. Wikipedia provides you some guidance.
What's important is that you define/use a set of terms that everyone in your team agrees with and uses. I believe that the terms you're proposing (static vs runtime) are not good because all these rules are exercised at runtime anyway. I would propose something like intrinsic vs extrinsic or internal vs external validation.
Why is it that higher level languages (Javascript, PHP, etc.) don't offer data structures such as linked lists, queues, binary trees, etc. as part of their standard library? Is it for historical/practical/cultural reasons or is there something more fundamental that I'm missing.
linked lists
You can implement a linked list fairly easily in most dynamic languages, but they aren't that useful. For most cases, dynamic arrays (which most dynamic languages have built-in support for) are a better fit: they have better usage and cache coherence, better performance on indexed lookup, and decent performance on insert and delete. At the application level, there aren't many use cases where you really need a linked list.
queues
Easily implemented using a dynamic array.
binary trees
Binary trees are easily implemented in most dynamic languages. Binary search trees are, as ever, a chore to implement, but rarely needed. Hashtables will give you about the same performance and often better for many use cases. Most dynamic languages provide a built-in hashtable (or dictionary, or map, or table, or whatever you want to call it).
It's definitely important to know these foundational data structures, and if you're writing low-level code, you'll find yourself using them. At the application level where most dynamic languages are used though, hashtables and dynamic arrays really do cover 95% of your data structure needs.
There are many definitions of "high level" when it comes to languages, but I will take it in this context to refer to languages that are purpose-built for a specific domain (4GL anyone?). Such "specific domains" are typically restricted in scope, for example: web page construction, report writing, database querying, etc. Within that limited scope, there is frequently little need for anything but the most basic data structures.
Is There a Need?
Let's consider the case of Javascript. The scope of this language was originally very bounded, being a scripting language than ran within the confines of a web browser. It was concerned primarily with providing a small amount of dynamic behaviour on otherwise static web pages. Furthermore, the limitations of the technology made it impractical to write large components in that environment (notably performance and the sandbox model).
Since Javascript was confined to address "small problems", there was little need for a rich set of data structures. As data structures go, the Javascript's map is very flexible. You must remember that Basic and FORTRAN went a long way providing only arrays -- maps are considerably more flexible than that. Javascript appears to be undergoing a transformation, escaping the sandbox. Some very ambitious systems are being built in Javascript, both within and outside of the browser. And the technology is advancing to keep up with it (witness the new Javascript engines, persistence models, and so on). I anticipate that the demand for more interesting data structures will increase, and that the demand will be met.
Library capabilities generally appear as needs arise. Many of the basic data structures are so easy to implement that it hardly seems worth adding them to a library -- especially if that library needs to go through some sort of standardization process. This is why so many languages (of all levels) do not provide them out of the box. But I think that there is another force at work that will change all that... the rise of multiprogramming.
A New Need Arising?
It wasn't too long ago that the code that most developers wrote ran within the confines of a single thread. But now, our systems are full of threads, web workers, agents, coroutines, clusters, clouds and all manner of concurrent systems. This changes the whole complexion of implementing data structures from scratch.
In a single-threaded context, it is trivial to implement a linked list in almost any language. But add concurrency to the mix and now it takes a great deal of effort to get it right. One really needs to be a specialist to stand a chance at all. That is why you see rich collection frameworks in all the latest languages. The need to share data structures across thread boundaries (or worse) is being fulfilled.
History... But Not The Future
So, to summarize, I think the reason why rich data structures are conspicuously absent from many languages is largely historical. The need was not great enough to justify the effort. But there are new forces at work, in the form of highly concurrent systems, that force language libraries to provide industrial strength implementations of the richer data structures.
My intuitive answer would be that these languages defer the higher-level data structures to the programmer to implement him/herself. This allows the programmers to custom tailor the specific data structure to the problem being solved by the software. Often in an organization, many of these DSes are packaged in libraries for re-use in a large-scale application.
Many programming languages share generic and even fairly universal features. For example, if you compared Java, VB6, .NET, PHP, Python, then you would find common functions such as control structures, numeric and string manipulation, etc.
What has been done to define these features at a meta-language (or language-agnostic) level?
UML offers a descriptive reference of software in every aspect, but the real-world focus seems to be data processes. Is UML relevant?
I'm not asking "Why we don't have a single language that replaces the current plethora." We need many different tools (at least in this eon).
I'm not asking that all languages fit a template -- assembly vs. compiled languages are different enough to make that unfeasible (and some folks call HTML a language, though I wouldn't). Any attempt would start with a properly narrow scope. In line with this, I wouldn't expect the model to cover even a small selection with full validity.
I would expect however that such a model could be used to transpose from one language to another (with limited goals -- think jist translation).
There have been many attempts at this, but none have been very successful. The earliest I'm aware of is UNCOL more than 50 years ago.
You've given a list of languages that have a lot in common because they're pretty similar -- they're all procedural languages with common roots and some OO extensions thrown in, so that's not too suprising. If you start looking at different languages like LISP, haskell, erlang, prolog, or even SQL you start seeing very different things.
What you're describing sounds like the formal semantics of programming languages. There are a variety of approaches and each will give a way to formally specify the meaning of a program in some programming language. In some cases, this specification is essentially a translation into another language such as lambda calculus, or compilation for a formally specified abstract machine such as SECD.
There is so much work here it's hard to pick a specific reference. But I hope I've given you some useful keywords to continue your search.
UML is typically used to define algorithms/code in simpler terms before moving on to real code.
To answer what I am guessing to be your question, there is already a defined set of required parts of languages while,for,if,else... Will this ever be set as a standard, or made into a base library that is used by all languages: no, this is because the different developers of languages like to do it themselves.
I think the closest you can get to this without loss of generality is a Turing machine, which is not very useful for practical purposes. But if you allow Turing machine languages to be "labeled" and reused, you could build up the concepts you need, working from low- to high-level.
I think that MOF is the universal language.
You can for example create UML diagrams from MOF via a UML metamodel. If you save this metamodel information into xmi then you can save what ever information you need and even more than in any language. XMI semantic is so rich that there is no limit to its use. If you map UML to xmi on the top of a metamodel live synchronize with MOF then this is for me the universal language.
The author of Pattern Calculus seems to propose such a universal model. I expect that it will turn out to be just as useful as previous attempts to define a universal model, that is to say, good in parts but not the last word.
I took a glimpse on Hoare Logic in college. What we did was really simple. Most of what I did was proving the correctness of simple programs consisting of while loops, if statements, and sequence of instructions, but nothing more. These methods seem very useful!
Are formal methods used in industry widely?
Are these methods used to prove mission-critical software?
Well, Sir Tony Hoare joined Microsoft Research about 10 years ago, and one of the things he started was a formal verification of the Windows NT kernel. Indeed, this was one of the reasons for the long delay of Windows Vista: starting with Vista, large parts of the kernel are actually formally verified wrt. to certain properties like absence of deadlocks, absence of information leaks etc.
This is certainly not typical, but it is probably the single most important application of formal program verification, in terms of its impact (after all, almost every human being is in some way, shape or form affected by a computer running Windows).
This is a question close to my heart (I'm a researcher in Software Verification using formal logics), so you'll probably not be surprised when I say I think these techniques have a useful place, and are not yet used enough in the industry.
There are many levels of "formal methods", so I'll assume you mean those resting on a rigourous mathematical basis (as opposed to, say, following some 6-Sigma style process). Some types of formal methods have had great success - type systems being one example. Static analysis tools based on data flow analysis are also popular, model checking is almost ubiquitous in hardware design, and computational models like Pi-Calculus and CCS seem to be inspiring some real change in practical language design for concurrency. Termination analysis is one that's had a lot of press recently - The SDV project at Microsoft and work by Byron Cook are recent examples of research/practice crossover in formal methods.
Hoare Reasoning has not, so far, made great inroads in the industry - this is for more reasons than I can list, but I suspect is mostly around the complexity of writing then proving specifications for real programs (they tend to get big, and fail to express properties of many real world environments). Various sub-fields in this type of reasoning are now making big inroads into these problems - Separation Logic being one.
This is partially the nature of ongoing (hard) research. But I must confess that we, as theorists, have entirely failed to educate the industry on why our techniques are useful, to keep them relevant to industry needs, and to make them approachable to software developers. At some level, that's not our problem - we're researchers, often mathematicians, and practical usage is not foremost in our minds. Also, the techniques being developed are often too embryonic for use in large scale systems - we work on small programs, on simplified systems, get the math working, and move on. I don't much buy these excuses though - we should be more active in pushing our ideas, and getting a feedback loop between the industry and our work (one of the main reasons I went back to research).
It's probably a good idea for me to resurrect my weblog, and make some more posts on this stuff...
I cannot comment much on mission-critical software, although I know that the avionics industry uses a wide variety of techniques to validate software, including Hoare-style methods.
Formal methods have suffered because early advocates like Edsger Dijkstra insisted that they ought to be used everywhere. Neither the formalisms nor the software support were up to the job. More sensible advocates believe that these methods should be used on problems that are hard. They are not widely used in industry, but adoption is increasing. Probably the greatest inroads have been in the use of formal methods to check safety properties of software. Some of my favorite examples are the SPIN model checker and George Necula's proof-carrying code.
Moving away from practice and into research, Microsoft's Singularity operating-system project is about using formal methods to provide safety guarantees that ordinarily require hardware support. This in turn leads to faster performance and stronger guarantees. For example, in singularity they have proved that if a third-party device driver is allowed into the system (which means basic verification conditions have been proved), then it cannot possibly bring down that whole OS–he worst it can do is hose its own device.
Formal methods are not yet widely used in industry, but they are more widely used than they were 20 years ago, and 20 years from now they will be more widely used still. So you are future-proofed :-)
Yes, they are used, but not widely in all areas. There are more methods than just hoare logic, some are used more, some less, depending on suitability for given task. The common problem is that sofware is biiiiiiig and verifying that all of it is correct is still too hard a problem.
For example the theorem-prover (a software that aids humans in proving program correctness) ACL2 has been used to prove that a certain floating-point processing unit does not have a certain type of bug. It was a big task, so this technique is not too common.
Model checking, another kind of formal verification, is used rather widely nowadays, for example Microsoft provides a type of model checker in the driver development kit and it can be used to verify the driver for a set of common bugs. Model checkers are also often used in verifying hardware circuits.
Rigorous testing can be also thought of as formal verification - there are some formal specifications of which paths of program should be tested and so on.
"Are formal methods used in industry?"
Yes.
The assert statement in many programming languages is related to formal methods for verifying a program.
"Are formal methods used in industry widely ?"
No.
"Are these methods used to prove mission-critical software ?"
Sometimes. More often, they're used to prove that the software is secure. More formally, they're used to prove certain security-related assertions about the software.
There are two different approaches to formal methods in the industry.
One approach is to change the development process completely. The Z notation and the B method that were mentioned are in this first category. B was applied to the development of the driverless subway line 14 in Paris (if you get a chance, climb in the front wagon. It's not often that you get a chance to see the rails in front of you).
Another, more incremental, approach is to preserve the existing development and verification processes and to replace only one of the verification tasks at a time by a new method. This is very attractive but it means developing static analysis tools for exiting, used languages that are often not easy to analyse (because they were not designed to be).
If you go to (for instance)
http://dblp.uni-trier.de/db/indices/a-tree/d/Delmas:David.html
(sorry, only one hyperlink allowed for new users :( )
you will find instances of practical applications of formal methods to the verification of C programs (with static analyzers Astrée, Caveat, Fluctuat, Frama-C) and binary code (with tools from AbsInt GmbH).
By the way, since you mentioned Hoare Logic, in the above list of tools, only Caveat is based on Hoare logic (and Frama-C has a Hoare logic plug-in). The others rely on abstract interpretation, a different technique with a more automatic approach.
My area of expertise is the use of formal methods for static code analysis to show that software is free of run-time errors. This is implemented using a formal methods technique known "abstract interpretation". The technique essentially enables you to prove certain atributes of a s/w program. E.g. prove that a+b will not overflow or x/(x-y) will not result in a divide by zero. An example static analysis tool that uses this technique is Polyspace.
With respect to your question: "Are formal methods used in industry widely?" and "Are these methods used to prove mission-critical software?"
The answer is yes. This opinion is based on my experience and supporting the Polyspace tool for industries that rely on the use of embedded software to control safety critical systems such as electronic throttle in an automobile, braking system for a train, jet engine controller, drug delivery infusion pump, etc. These industries do indeed use these types of formal methods tools.
I don't believe all 100% of these industry segments are using these tools, but the use is increasing. My opinion is that the Aerospace and Automotive industries lead with the Medical Device industry quickly ramping up use.
Polyspace is a a (hideously expensive, but very good) commercial product based on program verification. It's fairly pragmatic, in that it scales up from 'enhanced unit testing that will probably find some bugs' to 'the next three years of your life will be spent showing these 10 files have zero defects'.
It is based more on negative verification ('this program won't corrupt your stack') instead positive verification ('this program will do precisely what these 50 pages of equations say it will').
To add to Jorg's answer, here's an interview with Tony Hoare. The tools Jorg's referring to, I think, are PREfast and PREfix. See here for more information.
Besides of other more procedural approaches, Hoare logic was in the basis of Design by Contract, introduced as an object oriented technique by Bertrand Meyer in Eiffel (see Meyer's article of 1992, page 4). While Design by Contract is not the same as formal verification methods (for one thing, DbC doesn't prove anything until the software is executed), in my opinion it provides a more practical use.