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Scan and reduce/fold as generator expressions and list/set comprehensions that can access the previous iteration output.

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Would you like to replace this:

reduce(lambda prev, x: abs(prev - x), [2, 3, 4, 5])

by this?

last(abs(prev - x) for x in [2, 3, 4, 5])

Why not?

>>> from functools import reduce
>>> from pyscanprev import enable_scan, last
>>> @enable_scan("prev")
... def evaluate(data):
...     return reduce(lambda prev, x: abs(prev - x), data), \
...            last(abs(prev - x) for x in data)
>>> evaluate([2, 3, 4, 5])
(2, 2)


Example Description
Comparison Simple scan/accumulate and fold/reduce examples using the PyScanPrev resources, the Python standard library and the 3-for-section comprehensions for comparing readability and expressiveness.
Conditional Toggling Sometimes feedback isn’t really required, but you should at least store some state about what’s going on in the input. That’s the case in this example, which toggles/updates the state only for certain inputs, aLtErNaTiNg CaSeS, iGnOrInG oThEr ChArAcTeRs. Historically, PyScanPrev was created after thinking on how would be the simplest way to solve the problem described in this example.
Examples from the itertools.accumulate documentation

This is a copy and an adaptation of the itertools.accumulate examples in the Python documentation to use the PyScanPrev comprehensions instead of that standard library function. The examples include:

  • running product
  • running maximum
  • amortization tables
  • chaotic logistic map
Factorial and Multiplication Fold/reduce factorial and multiplication / product of a sequence implementation using PyScanPrev.
Fibonacci Fibonacci sequence with PyScanPrev.
Gray Code Generating Gray codes using the definition is slower than using bitwise operations, but the recursive definition can be written as a scan/fold expression and is useful for testing, as this example shows.
Single pole lowpass IIR Filter DSP (Digital Signal Processing) applications ofter requires feedback, i.e., accessing some previous output value in a process. PyScanPrev can be used to write simple signal processing models like IIR (Infinite Impuse Response) linear filters. This is an example with a single pole lowpass digital filter. For testing and displaying the results, this example uses AudioLazy and hipsterplot.
State-space model

Simulation of linear discrete time modeling in control engineering. The example gives an implementation of both time varying and time invariant state-space models. These models are then used to simulate:

  • an accumulator
  • a LTI (Linear Time Invariant) continuous time mass-spring-damper SHM (Simple Harmonic Motion) model
  • a linear time varying “leaking bucket”-spring-damper model
  • another LTI IIR filter, designed from the difference equation and simulated as a state-space model, compared with the AudioLazy results

The discretization process is included in the example, and the simulations use hipsterplot to plot the motion path/trajectory. This example includes an explanation/proof to the conversion from difference equations to a state-space model (via Z Transform).

Why “Scan”?

Scan is a high order function that can be seen as something between the map and the reduce (fold) functions, returning all steps of a fold. Since Python 3.3, it’s available in the function itertools.accumulate.

The goal here is to find an easy way to read the previous value in a generator expression and anything alike, so that the scan/fold (accumulate/reduce) code can be written using them. Readability counts!

Package contents


Decorator that allows functions to have generator expressions and list/set comprehensions with a variable (the one with the given name) in its body for accessing the previous resulting value.


Gets the last value from an iterable, making it straightforward to write a reduce/fold from the scan result.

prepend(value, iterable)

A version of [value] + some_list for general iterables, returning a generator. This function was created to allow PyScanPrev-enabled generator expressions and list/set comprehensions to include an explicit start value, but it can be used to prepend a value in any context, e.g. to force a start value on itertools.accumulate.

scan(func, iterable, [start], *, echo_start=True)

It’s an implementation of the scan higher order function with more features than itertools.accumulate (the start and the keyword-only echo_start parameters) and consistent to the functools.reduce function signature.

Tell me, how is that possible at all?

Magic! Some people say that’s bytecode manipulation, but isn’t that all the same?


You can either use pip:

# From PyPI
pip install pyscanprev

# From GitHub master branch
pip install --upgrade git+

Or directly:

python3 install

This software depends on bytecode, which requires Python 3.4+.

The world without this package (rationale)

It’s not usual nor widely known that the cross/cartesian product applied on multiple “for” sections in a generator expression or a list/set/dict comprehension allows more than one section to have the same target variable name. But that provides the means to do something akin to a scan, for example this cumulative sum (Tested in Python 2.6+ and 3.2+):

>>> [prev for prev in [0] for el in range(5) for prev in [prev + el]]
[0, 1, 3, 6, 10]

Whose parts are:

[prev for prev in [start]
      for target in iterable
      for prev in [func(prev, target)]]

But that’s a kludge, it’s hard to grasp, hard to change/update/maintain, fixed/locked in that “for” section order, and its behavior has some minor details whose control would need to be external (e.g. using the first value from the iterable as the start). The prev variable appears at least 4x in such structure and twice as a target. The first prev value is start, which is just seen/used by the last “for” section in its first func call, whose result is assigned to prev before the whole list comprehension appends/”yields” any output/result, since it’s also the target variable name in that “for” section. So start is never an output, although everything starts with prev for prev in [start].

It’s not only about code aesthetics or readability, but also about pattern memorization: knowledge about the scan abstraction and about the Python language is probably not enough for one to remember that structure.

As func in the previous example was essentially operator.add, let’s do the same cumulative sum with itertools.accumulate (Python 3.2+):

>>> from itertools import accumulate
>>> list(accumulate(range(5)))
[0, 1, 3, 6, 10]

It seems the same, but here the first zero output is the next(iter(range(5))), not the result of a sum or any other func for that matter (i.e., it doesn’t depend on func at all). To be really equivalent to the 3-for-sections list comprehension above, it would need to be something like:

>>> list(accumulate([0, 0, 1, 2, 3, 4]))[1:]
[0, 1, 3, 6, 10]

We had to prepend 0 to range(5). What’s going on here is that accumulate returns a generator that yields the values:

[i0, i0+i1, i0+i1+i2, i0+i1+i2+i3, i0+i1+i2+i3+i4, ...]

Where “in” is the n-th value from the iterable input. Every step obviously re-uses the previous step result instead of summing all the previous inputs again, and that’s what the scan is all about. On the other hand, the 3-for-sections list comprehension does this when func is the sum/add:

[s+i0, s+i0+i1, s+i0+i1+i2, s+i0+i1+i2+i3, s+i0+i1+i2+i3+i4, ...]

Where “s” is the start. Since Python 3.3, itertools.accumulate has an optional second parameter, which should be a binary operator/function/callable. For a given func, the resulting generator would yield, in order:

next(iterable),                  # result[0]
func(result[0], next(iterable)), # result[1]
func(result[1], next(iterable)), # result[2]
func(result[2], next(iterable)), # result[3]

Where start is implicit as the first value from iterable, and result is that output iterable itself seen as a sequence. To grasp the difference, let’s see a cumulative sum of squares starting with 3 in the accumulator/register.

>>> list(accumulate([3, 5, 1, 1, 2], lambda x, y: x + y ** 2))
[3, 28, 29, 30, 34]

To get the same result with a list comprehension, one would do:

>>> [3] + [x for x in [3]
...          for y in [5, 1, 1, 2]
...          for x in [x + y ** 2]]
[3, 28, 29, 30, 34]

There’s also a really old package in PyPI called functional, whose last update was in 2006. Besides the distinction between non-strict and “prime”/strict counterparts, it mimics all the 4 scan and 4 fold Haskell functions, including their names and their parameter order. The functional.scanl1 and the itertools.accumulate functions are almost the same, the difference is that scanl1 needs the function to be the first argument and it isn’t optional. On the other hand, functional.scanl needs an extra “start” parameter. Both functions return a generator:

>>> import functional, operator

>>> # scanl (+) 0 [0..4]
>>> list(functional.scanl(operator.add, 0, range(5)))
[0, 0, 1, 3, 6, 10]

>>> # scanl1 (+) [0..4]
>>> list(functional.scanl1(operator.add, range(5)))
[0, 1, 3, 6, 10]

>>> # scanl1 (\x y -> x + y^2) [3, 5, 1, 1, 2]
>>> list(functional.scanl1(lambda x, y: x + y ** 2, [3, 5, 1, 1, 2]))
[3, 28, 29, 30, 34]

Both scanl and scanl1 have a behavior different from that 3-for-sections list comprehension.

Python functools.reduce, functional.foldl and functional.foldl1, as fold/reduce implementations, share a core idea: they return the last value of the scan resulting from the same given inputs to functional.scanl and functional.scanl1. The reduce function can have an optional start as the 3rd and last argument, which gives to it both the behavior of both foldl, that requires the start as the 2nd parameter, and foldl1, which uses the first iterable value as the start value. If there’s a way to modify generator expressions so that scanl/scanl1/accumulate can be implemented with them with a good readability, the same would apply to reduce.

But, even for developers who like to think on these concepts as ready to use abstractions stored in first class objects, here we got a parameter hell! Their order is a mess:

  • (iterable, func) -> itertools.accumulate
  • (func, start, iterable) -> functional.scanl
  • (func, iterable) -> functional.scanl1, map, filter
  • (func, iterable, [start]) -> functools.reduce

The higher-order functions scan and fold appears respectively in itertools.accumulate and functools.reduce first-class objects (functions are first-class objects in Python), which are quite easy for people coming from a functional programming background to grasp, and far easier to read/remember than the 3-for-sections list comprehension. One just neet to know these two have their 2 parameters reversed, and that accumulate doesn’t have an optional external start value. It would be great to have an optional start parameter on itertools.accumulate, as well as a function signature standardization, but the main purpose of this is just to get a cleaner alternative to that 3-for-sections list comprehension.

Copyright (C) 2016 Danilo de Jesus da Silva Bellini

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