1. talk at Leipzig's Haskell conference HaL-10

    As I said before: I will hold a small talk about dogelang at Leipzig's next Haskell conference HaL-10.

    Here's a translation of the abstract (original in German):

    dg - The better Haskell is named Python

    Who needs typeclasses? Or static type system even? Why have side-effects to be so complicated? And what are monads?

    There must be something better!

    dg is Python with Haskell syntax. In a short talk (15 minutes) I want to introduce, on a funny note, the "better Haskell".

    Under the slogan "what we don't have, you won't need" we will be looking at features of the Haskell language and their counterparts in dg. To conclude, we will take a look at how intelligent a Python program can look, if it has the proper syntax.

  2. Published: Thu 12 November 2015

    Deciding on the pure-ness of a python function

    Through the work with dogelang I am currently looking at pureness of python functions and on how to decide and annotate pureness so that I, in theory, could write a static checker that checks whether the annotated functions are pure.

    We can take the definition of pureness from the wikipedia and it is as follows

    1. The function always evaluates the same result value given the same argument value(s). The function result value cannot depend on any hidden information or state that may change while program execution proceeds or between different executions of the program, nor can it depend on any external input from I/O devices.
    2. Evaluation of the result does not cause any semantically observable side effect or output, such as mutation of mutable objects or output to I/O devices.

    Python makes few restrictions on side-effects and mutability and since we even can override basic operators (like the dot-operator for attribute access), it is hard to infer anything remotely useful or non-trivial from a python-program.

    I have been, for example, responsible for code like that (paracoded for your convenience):

    def cls(object):
      def __init__(self):
        self.d = {}
      def __getattr__(self, attr)
        if not attr in self.d:
          self.d[attr] = someobj(attr)
        return self.d[attr]
    A = cls()
    A.foo # this line has side effects

    This violates the basic principle of python that an expression should either have a return value or a side-effect but not both. But we might have done this for good reason. In this case, the academic value of the "front-end" of the library outweighs the atrocities of the implementation. But it serves as a fine example that, it is hard to reason about even simple looking expressions, if they include the dot-operator (or any other operator for that matter), especially if you don't have the full class definition at hand.

    But it isn't only user-made classes that don't hold true to that principle. Most low-level functions that interact with the operating system do that in some way. They have their side-effect within the realms of the rest of the system and still they have a return value that represents some aspect of the operation within our program.

    os.write(fd, str)
    Write the bytestring in str to file descriptor fd. Return the number of bytes actually written.
    open(file, ...)
    Open file and return a corresponding file object. If the file cannot be opened, an OSError is raised.

    So we would want to mark any function as impure that has a reference to one of these impure functions. But we would also want to mark any functions defined within these impure functions to be themselves impure. Looking at the following code it would be hard to infere whether a function has a reference to, let's say 'open'.

    o, f = open, "file.tmp"
    def fun():

    Given Rice's theorem, I'd think that properly tracking references would either be undecidable or at least a problem of exponential growth (tracking all possible assignments to some reference). So to be on the safe side, we state the following as our impurity rule number 1: A function is impure if it uses a reference from an impure function or has a reference to an impure function. We would also want an exhaustive list of builtin impure functions, applying the rules recursively.

    In the former code example the surrounding block leaks (that's what we will call it) the reference to both o and f into the namespace of fun. This would not be a problem. We even want this behaviour. Let's look at this example:

    def f():
      return x*x
    x = 5
    print(f()) # 25
    x = 7
    print(f()) # 49

    This is contrary to our definition of purity, that the function, given the same arguments should evaluate to the same value. So this only happened, because the reference was updated. Let's add impurity rule number 2: A function is impure if a name is assigned twice and a reference to that name is leaked. This should cover all cases since shenanigans like the following cause a NameError.

    def a():
      yield (lambda : x*x)
      x = 3
      yield None
    x, i = 5, a()
    g = next(i)
    print(g()) # Name Error: free variable 'x' referenced before assignment in enclosing scope

    Here we see the yield-keyword we have the case of suspended computation. The next-function is of course part of the set of impure builtin functions because it returns a value and advances the internal state of the generator. Since for rule 2 we introduced the leaked name, we can still allow multiple assignments within a function if the name is not leaked because we want to allow, for example, the generator-function infinity to be pure.

    def infinity():
      i = 0
      while True:
        yield i
        i = i + 1

    Here, I find, we arrive at an impasse. Every iteration over a generator "uses the generator up". We could say that we won't allow references to generator-objects but that seems arbitrarily strict. Also, we can't really know which name could be used by a generator-object which would lead to not allowing any bare names (only function calls). Disallowing generators seems impossible since map and filter and other functional primitives depend on it.

    I would welcome any ideas for a rule that would solve this issue of leaking the internal state of a generator.

  3. Musings on Graph Isomorphism pre-results

    I think by now everybody has read that Babai et al will hold a seminar next week where they want to "outline an algorithm that solves the Graph Isomorphism problem [...] in quasipolynomial time".

    So they are shooting for a result with the complexity \(2^{O(log(n))^c}\) while the best former result (by Luks, 1983) for Graph Isomorphism was \(2^{O(\sqrt{n*log(n)})}\).

    What is Graph Isomorphism? Given two graphs, the question is, whether there is a mapping function \(f\) between the nodes of one graph \(G_1\) to the nodes of the other graph \(G_2\) so that every edge \((a,b) \in G_1\) is also an edge \((f(a),f(b)) \in G_2\).

    So why is this quasipolynomial when \(2^x\) is obviously not a polynome? That's where the quasi comes from. If the \(c\) in \(2^{O(log(n))^c}\) is \(1\) then this simplifies to \(2^{O(log(n))}\) which is a polynomial. Remember that the base of both the logarithm and the exponentiation isn't actually \(2\) so this doesn't simplify to \(O(n)\) but some other polynomial (but a polynomial nevertheless). And the \(c\) is just a factor in the formula (like the base the logarithm and the exponentiation) that have no impact on the complexity properties (the general shape of the formula describing the complexity) but can be changed by optimization (opposite to groundbreaking research). So given proper optimization the algorithm we will be presented could be (theoretically) as fast as a polynomial algorithm.

    When we talk about "fast" we talk about how the runtime changes (relatively) when the problem size grows larger. A "faster" algorithm has smaller relative growth while it might still be orders of magnitude slower in actual runtime for any practically sized input.

    Is this result surprising? Well, yes and no. On the one hand the "witness" (or positive solution to the problem) is only of linear size (one mapping for every node in the graph) and the algorithm for checking would be basically quadratic in complexity (checking the definition given above for all edges and there are only quadratically many). This holds for problems in NP-complete but also for problems in P. I think it is obvious that this answer would not be very hard to find (we surely don't need to check every possible mapping - this would basically be the case for NP-complete) but it isn't obvious that the answer is easy to find either (like for a problem in P).

    So what we're probably going to see is a clever application of some result of group theory that is applicable here and excludes a large number of all possible isomorphism as possible solutions. So the problem is still not easy but math can help a lot.

    What does it mean? It is not known whether Graph Isomorphism is NP-Complete. This result would be a strong hint, that it isn't (albeit no proof). Even if we found Graph Isomorphism to be in P, this would have no larger impact on the field of complexity theory in general.

    What's the next thing? That would be finding an algorithm properly in P. If such an algorithm exists, we would also find a way to compute a small "witness" (or in this case "proof trail") that is an easily checkable answer to the question "is there no such isomorphism?". How this proof trail would look is even less obvious. There could be some function that, given a graph as input calculates a value and all graphs that have an isomorphism between them have the same value (which would also need to not be exponential in size). In practice it is very difficult to even find hard instances of Graph Isomorphisms. So it might turn out that there are actually only easy cases, we just don't know it yet.

    Read more and differently here:

    If it wasn't obvious, I don't believe that P=NP because I don't believe that NP is closed under complement.

  4. dogelang adding pattern matching

    If you haven't read about dogelang, you really should. It is python with haskell-syntax where you can use all python modules. So basically coffee-script but for a language that is already useful. And python bytecode is emitted instead of first compiling to python itself, but that just as a side-node.

    On my behest, the developer of dg added pattern matching so that in an if-condition a tuple-unpacking-expression does not raise an exception if the unpacking fails but returns false.

    Therefore one can now implement their own zip and list-constructor in the following manner:

    zip = x' y' -> if
      ([x, *xs], [y, *ys] = x', y') =>
        yield (x, y)
        yield from $ zip xs ys
      otherwise => yield from ()
    list = x' -> if
      ([] = x') => []
      ([x, *xs] = x') => [x] + list xs

    This looks almost like haskell's case-pattern-matching but not quite, so it is sure to annoy every haskell-programmer.

    You might need to execute python3 -m dg -b once after installing/updating dg to rebuild the interpreter/compiler-bundle with this feature.

    In other news: It is quite likely that I will hold a small talk about dogelang at Leipzig's next Haskell conference HaL-10.

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