Reversible contextual Python with strong properties

I previously mentioned my effort to learn how best to do semi-formal development in Python. Here I report my progress, done during my end of year 2021 holiday break. 

The effort is to find a 'best design' to achieve certain properties, and I am doing this by rewriting code again and again to progress a 'design' towards a 'better design'. At rewrite 21 we have:

• A systematic way to write reversible, context aware code.
• Context specific algorithms can be defined over containers as well as over higher order functional applications.
• Logical and declarative 'free-variable' oriented style is possible.
• Function applications with context can be 'saved' to state and to be continued later or elsewhere.
• Quoted expressions can help define equalities and inclusion properties.

Some insights:

• This could be Java or C++! Python is handy with a focus on 'abstract' arguments, but traits would carry us through an OO implementations.
• What about types? The question is better expressed as: what about types within a formally strong stack? Types are good, but they too must be expressed 'with strength', otherwise they hinder you.
• How much value is there in the strong typing of Haskell and Scala? Much if you use it for formal properties. However if most of your types are 'structural boilerplate', then you are not actually gaining much. Formal properties with types means 'type level' programming, and that is not cheap, nor easy to scale. 
• This code does 'nothing real'!? It is not meant to! Its purpose is to show how design cost effective code with formal properties in 2022!

Here is a quick run through my efforts. To note each of these efforts are a 'throw' away effort with the goal of learning how to write 'stronger' more 'formal' code. Twenty-one rewrites later (about sixty hours of work), the effort was worth it:

1. This effort starts with a hard coded functions to swap a positional argument to keyword arguments and vice versa (the other way around).  Swapping is reversable, and our goal here is to write reversible code. 
• I built this 'keyword argument' constructor/destructor initially to bridge between the 'product' type nature of keyword arguments and the 'sum' property of positional arguments. Later, I realised that keyword are named, positional arguments are not (conceptually!), and that this first focus allows me to move freely between a nameless point-free style of programming with positional args, to a more classical 'named' style. This explains why I am still point-free when I use identifiers, because these can be removed with well chosen operators.
2. A generic keyword argument' constructor/destructor function can be implemented by simply 'accessing kwargs'. However, Python has no efficient ways to remove keys from a dict, and that forces us to use 'ugly' copy constructs to 'pass on' a kwargs, minus the chosen keyword. Therefore  version two uses eval and lambda to generate a an effcient generic keyword argument' constructor/destructor. 
• Retrospectively, I would have avoided the eval, and stuck with kwargs and copying the dict to remove an arg.
3. Lambdas are a pain to debug and to type annotate,  version three uses callable objects instead of lambdas.
4. None of the above is composable nor 'stateful' (from a 'traditional' perspective), as all happens within function arguments. Version 4 adds function composition and introduce a AKW class, to 'statefully' hold function arguments. It is 'reversible', in that we can 'call' (cocall) a function with the object's data as arguments. 
• AKW stands for Args KWargs, as it holds both the args tuple and the kwargs dictionary.
5. This is all dynamic typing, and initially at least, with no type conditional expression. However, to 'feel the ground', version 5 does the following: if two complementary functions/lambdas are expressed as callable object, then we can give their type classes a class attribute which programmatically provides this complementary tie. For example, given d_k1() an operator from keyword arg k1 to posional arg, one can find its complementary operator c_k1(), from a positional arg to a keyword arg k1.
• This works, we do not need it (yet).  
6. Focusing on function arguments is somewhat 'stream oriented'. The AKW class introduced in 4, is more of a classical 'stateful' approach. Version 6 shows how function composition can be constructed on this stateful view.
7. Functions and callable objects are nice. Yet by default they force us to create 'return value states'. These return states 'semantics' are different and do not naturally fit within our simple 'call' semantics. Another aspect is they are 'invisible' to the notion of reversible code.  Therefore version seven takes a first go at using tail contination to avoid 'return states' of functions. 
8. Continuation passing can express a 'deeper' execution context with the help of a 'stack' of tail continuations. Version 8 introduces a continuation 'list' for this purpose.
• I believe Gerald Jay Sussman's keynote and discussion - reClojure 2021 mentions using a stack of tail continuations in his recent work. I don't have much scheme/lisp practical experience, and therefor do not know how much stacking tail continuations 'is common'. 
9. Version 9 extends operators to have something equivalent to the notion 'quoted' expressions in Scheme/Lisp.
• While not futher developed here, 'quoting' is an 'extra dimension' and extremly valuable to define notions of equalities, and of inclusion (e.g. conbined with matching).
10. Version 10 introduces a basic map function. 
• Implemented for tuple, list, sets and dicts, the mapping is not always reversible. Which highlights that certain operations need more support to guarantee properties of reversibility. For example, mapping the set 1,2,3 to 1,1,1 is not reversible.
11. The classical tail continuation is only telling us where the output of a function should go. With a 'stack' / 'list' of continuations, we can carry more information, for example the 'index' or 'key' of the element that is currently being mapped. Version 11 shows how this is done, and shows how we can make a map able to recurse over hierchies of containers. Also this rewrite introduces a first 'partition' 'unpartion' operator.
12. Here we partition and unpartition classes of dataclass fields. 
• Object orientation is stronger when 'done backwards'. That is, composing general objects quickly breaks. However, given a large object that expresses a real world scenario, we can sometimes partition it 'with good formal results'.
13. Logical programming allows us to perform 'or' and 'and' like operations, and often have something similare to a 'not'. Version 13, shows how logical programming is 'easy' with tail continuation.
• Five class/functions are defined Success, Fail, First, All, and Fails. 
• Sucess always succeeds.
• Fail always fails.
• First is succeeds when the first of its arguments succeeds.
• All succeeds when all of its arguments succeed.
• Fails succeeds when its single argument fails, and fails when it succeeds.
• Here __call__ is success while 'fail' is called on failure. Later __call__ with the named keyword 'fail'=True indicates failure.
14. Adding a 'match' operator (with free-variable) to logical programming gives us a simple way to write declarative programs (here in Python).
•  The 'larger' Prolog program I wrote was a prototype compiler to assign 'topologically' senstive types (e.g. eges vs face vs element, etc.) and perform other transformations to a domain specific modeling language for semicondutor device simulations (applied math mid-90s).
• My current view about declarative programming (logical programming is a form for declarative programming): They only scales when embedded in a more general semantic stack (as done here for example).
15. Logical programming is more powerful with lambda expressions (or callable objects). Version 15 defines the operator SelectApplyN1, which apply multiple operators to a single argument, and keeps those that succeed. The goal is to validate that we can mix declarative and lambda like function application.
• Example: 
• Given: SelectApplyN1((Above(3), Above(5), Above(7), Above(8)))(6, continuations=...
• We get: (((7, 6), (8, 6)),) having defined an appropriate 'Above' operator.
16. In the previous code rewrites, spcial keyword arguments are defined (e.g. for continuation passing, for the free-variable list) and are explicitely defined in all the function signatures. Version 16, pushes this approach in a slightly non-sensical way by adding the keyword argument 'state' to the reference call signature, and explicitely adding this 'state' argument throughout the code. This results in the question: 'why be explicit'?
17. Version 17, only declares keyword arguments explicitly when used locally. However, the tail continuation list is an explicit keyword argument, and expressed everywhere. The 'standard' call signature looks like this:
• def __call__(self, *args, continuations: Optional['Continuations'],  **kwargs):
18. At which point I ask myself: "why not go all the way"? Therefore version 18 moves the continuation stack to a positional argument, as follows:
• def __call__(self, continuations: Optional['Continuations'],  *args, **kwargs):
• To note that we are back to a more 'classical' point-free formulation, as 'continuations' is positional and not keyword based.
19. Context annotation is revisited in version 19, and expressed more generically. To be specific, the continuation stack gets dynamically decorated with contextual information that can be queried 'within context'. These decorators are referred to by 'type'.
• The example code tags the logical operataors First and All (see 13), so that within the arguments of these functions can query 'which argument index are we'? To the example presented in 15 above, we get (((7, 6, 2), (8, 6, 3)),), because the successful operators were the third and fourth (starting from 0).
20. Continuing on context annotation, the recursive map of version 11 is adapted in version 20 and a 'depth' context specific function is introduced. The goal is to show that while "positional" context annotations is a handy nice, even more valuable is the ability to algorithmically extend your contexts.
21. Finally, we conclude this series with version 21 and a careful change ensures that the complete context stack is captured when capturing the state of compuation (e.g. with AKW as introduce in version 4).  This context becomes active again when we 'continue' compuations from each saved state (with cocall).

All original content copyright James Litsios, 2022.

Lowering your continuations while lifting your functions (in Python)

 A Previous post describes Python code that does functional composition on reversible code. A question that one asks when starting to write compositional code is: how does one compose functions 'at different levels'? And I purposely express this a bit ambiguously as 'different levels' can mean different things. Today we will care about levels within a hierarchy of data containers and simple objects, and note a thing or two on the way.

The map function is Python's builtin tool to apply a function over each data of a container. Combined with a lambda or a function definition we can convert fonctions on container elements to functions on containers. Jargon for this type of code is that we are lifting a function to the container. The idea being that a function on a container is higher than function on container elements.

An important detail to note, is that while the mapped function is being lifted, the continuation context needs to be 'lowered'. This is because the lifted function will later be called with a continuation context for the container, yet internally the original mapped function is called, and it will need 'additional' continuation context. Therefore an extra level of continuation is added in to 'lower it' down to the level of the mapped function. Two examples are given, the first  

• "Lowers down" a detached generic continuation context. 

python

The second

• "Lowers down" a location carrying continuation. 

python (container), python (object)

Another particularity of this mapping code is that these extended / lowered continuation are data constructors. Which makes sense because for each element we need its return value to be become data to be put into the resulting mapped container.  This data constructor nature of the continuation extension also reflects the nature of the continuation changing as it enters the data container that is being mapped.

And this brings up an anecdote:

As I wrote these mapping functions, I was immediately reminded of the Newton programming language by Prof. Rapin EPFL. Sadly, I could not find a freely available document on Newton, yet if my memory serves me well, Newton had a higher order container location abstraction that provided a service not very different to our subject here. (One of my CS MS semester projects had me extending the Newton compiler,  in ~1985).

This continuation 'extension' technic has strong ties to the zipper structures for container traversals.

Yet more importantly it reminds us that within a more formal approach to code function arguments may be expressing:

1. What is coming in to the function.
2. The call contexts to the function.
3. Contexts of immediate use of the function's results.

Within a continuation passing style, it is therefore perfectly ok to decorate your continuations to communicate more context to your functions. In the example code referred to here, the 'lowered' continuation carries index positions, dictionary keys, and dataclass field names. 

All original content copyright James Litsios, 2021.

Data composition from functional composition in Python

Previous post describes code that does functional composition. Let's now add operators that convert this functional composition to become data composition, and allow what is produced to be 'executed later'. Github drop for these code snippet here.

Key insights in writing code this way are:

• Data objects are derived from the functions which define their semantics.
• This is meta-programming 'done backwards'. Like with meta-programming we capture 'what could be interpreted', but we do so through the execution semantics, not by adding an additional meta-language.
• Data objects execution semantics can be ambiguous until ambiguities are removed. Not far from a "quantum-like" execution model!
• There are strong ties here to laziness and trampolining.
• Blockchains are as much about composition of data, as about composition of functions.

These example code snippets are very short (<100 active lines). Two design choices make this possible: 

1. Focus on arguments, not return value, nor object state
2. Make everything composable 

What these examples hint at is:

• Return values are really just arguments to further calls. 
• Object states are really just arguments, references (e.g. function reference), and contexts of possible further calls.

This is in fact true at the most simple level, such as in the code tied to this posting. 

All original content copyright James Litsios, 2021.

Point-free reversible Python

Have you ever wanted to to write 'formal style' in Python?

If so, these snippets of code which I shared on gitbub might please you.

What is a formal style of programming

Formal "code", like formalism in mathematics, is to solidly tie what you write to what you mathematically mean. And that connection goes both ways: from math to expression, from expression to math. 

The traditional way to write formal code is to prove as much as you can 'along the way' of writing code. The difficulty of this approach is that proofs are hard, which either limits the complexity of your code, or limits your ability to be productive.

Another way to write code with mathematical properties is to assemble pieces of code that already have mathematical properties. When this assembly preserves these mathematical properties, and even better when it scales them, the resulting code has mathematical properties.  A key concept here is that these "pieces of code with mathematical property" are "at different levels", as it is the interplay between these levels that give the code its semantic depth.  (This last point explains why languages like Prolog are hard to scale).

Dualities are good place to find mathematical properties.  An example of duality is symmetry: something can be 'one way' and 'the other', and we know "mathematically" what that means. This is why reversible code,  reversible types, adjoint structures, and pretty much anything that goes 'one way' and 'the other' (e.g. like rows and columns) are a good basis on which to build systems with formal properties.

What is reversible programming?

Reversible expressions can be evaluated 'one way' and 'another'. This might be execution evaluation that might be reversed, yet might also mean that the interpretation of the expression can be seen from a reversed perspective at different semantic level (e.g. types, external view).

Reversibility is also a way to approach multi-party programming: Say I am party Alice having received B from Bob, after having previously shared A with Bob. Alice can reverse compute B to A to validate that Bob computed B from A. That is a simplified view of things, but it captures the idea that reversibility enables trust.

Reversibility is much like immutability that has been weakened, but which has not lost its magical properties.

What is point-free style?

A point-free (or tacit) style of programming traditionally means coding by composition without naming arguments (and without assignments). In a Python centric view that could mean to only use 'positional arguments', and to use only Callable objects (so as not to have method names), but you need to ask yourself: "Is that important"? In fact within the scope of reversible programming, point-free is best approached as composable and reversible. That you name things or not is of lesser importance. In fact, it is of lesser importance also because the traditional point-free is to focus on 'expression space'. Yet modern programming happens much at the type level, or the meta-type level, and a strict point-free approach would lead us to writing code where types are nameless, as well as meta-types. That would be tricky!

Note here: Point-free much leads to array, stream and structural oriented programming. This is not lost here. In fact a main reason to write code like this in Python is to write structural oriented ML. Reversibility is also a vital piece of blockchain and smart contracts theory too.

Why should you care?

Wikipedia's programming paradigm page enumerates many ways to code. Reversibility is one of the concepts that brings together all of these different ways of expressing software. Putting reversibility 'underneath' a paradigm gives it bridges to other paradigms that have also been given reversibility properties. 

All original content copyright James Litsios, 2021.