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PCM20210731_SICP_1.2.1_LinearRecursionAndIteration
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### A Pluto.jl notebook ###
# v0.20.0
using Markdown
using InteractiveUtils
# ╔═╡ 38733a25-5f53-4ffe-8ae0-4650207fa9e3
begin
using Pluto
using Plots
using LaTeXStrings
using Statistics
using GraphRecipes
#----------------------------------------------------------------
println("pkgversion(Pluto) = ", pkgversion(Pluto))
println("pkgversion(Plots) = ", pkgversion(Plots))
println("pkgversion(LaTeXStrings) = ", pkgversion(LaTeXStrings))
println("pkgversion(Statistics) = ", pkgversion(Statistics))
println("pkgversion(GraphRecipes) = ", pkgversion(GraphRecipes))
end # begin
# ╔═╡ 45de6af0-f228-11eb-3398-0ba586081633
md"
===================================================================================
#### SICP: [1.2.1 Linear Recursion and Iteration](https://sarabander.github.io/sicp/html/1_002e2.xhtml#g_t1_002e2_002e1)
##### file: PCM20210731\_SICP\_1.2.1\_LinearRecursionAndIteration
##### Julia/Pluto.jl-code (1.11.2/0.20.3) by PCM *** 2025/01/07 ***
===================================================================================
"
# ╔═╡ c2876964-60ab-4ee9-80b6-4b4f884f360a
md"
##### 0. Introduction
*In this section we will examine some common 'shapes' for processes generated by simple procedures. We will also investigate the rates at which these processes consume the important computational resources of time and space. The procedures we will consider are very simple. Their role is like that played by test patterns in photography: as oversimplified prototypical patterns, rather than practical examples in their own right.* (SICP, 1996, 2016, p.41)
In our own words we want to deal with the relation between the *syntactical* form of the *code* and the *semantical* consequences in measures of *time* and *space complexity*.
"
# ╔═╡ 5884ee27-01f9-4a7e-8f3f-60a2ac3d5733
md"
---
##### 1. Topics
- linear *recursive* process, linear *iterative* process
- linear *recursive* syntactical form, linear *iterative* syntactical form
- mathematical *induction*
- big *O* notation, time complexity, space complexity
- *tail-recursive* optimization (tco)
- *base* function $isone$
- iterative *loop* constructs: $while, for$
- *abstract* type Signed, AbstractFloat, Integer
- multiple bindings: ... = ... = <value>
- shortcuts: +=, *=
- *Union* of types
"
# ╔═╡ 76dd1717-3458-492f-905f-733a5c5b8b8f
md"
---
##### 2. Libraries
"
# ╔═╡ 71a5c80b-d0c1-4269-88f3-d75cbc4ecce5
md"
---
##### 3. SICP-Scheme-like *functional* Julia
"
# ╔═╡ b7a1a820-167e-4509-9132-16d1e373a599
md"
###### 3.1 Mathematical *Specification* of *Factorial* Function
- ###### Linear *Recursive* Form:
$fac(n) = \cases{ 1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if } n=1 \cr n*(n-1)! \;\;\text{ if } n \ge 2}$
$\;$
$\;$
- ###### Linear *Iterative* Form:
$fac(n) = 1 \cdot 2 \cdot 3 \; \cdot \; ... \; \cdot \; (n-1) \cdot n = \prod_{i=1}^n i$
$\;$
$\;$
$\;$
The linear *recursive* form has the same structure as a proof by [*mathematical induction*](https://en.wikipedia.org/wiki/Mathematical_induction).
$\;$
"
# ╔═╡ f5869f82-5163-4ca9-8344-fbb5af822a0a
md"
###### 3.2 *Linear Recursive* and *Generic* Method of Function $factorial$
The term *linear recursive* stems from the fact that the demands in *space* and *run time* depend in a *linear* fashion from the parameter $n$.
Rates of growth in time and space (= *complexity*) can be described by the *big-O-notation*. *The idea of big-O notation is that a function $f$ is $O(g)$ if $g$ is an upper bound for $f$, for big enough inputs, ignoring constant factors.*
*Formally: $f$ is $O(g)$ if there are constants $c$ and $m$ such that $|f(n)| \le cg(n)$ for all $m \le n$* (Sanella et al., 2021, p.191).
Linear recursive *coded* functions always generate linear recursive *computational* processes with *O(n)* *time* and *space complexities*
$\;$
"
# ╔═╡ 76c277f3-f31f-4f6b-aae1-94fbbda60be4
function factorial(n)
==(n, 1) ?
1 :
*(n, factorial( -(n, 1)))
end # function
# ╔═╡ ab72cb48-9398-42e1-8ed7-8bc8db399ac5
factorial(6)
# ╔═╡ 362f76e2-6ba0-4564-90df-3c4258d88c83
factorial(6.)
# ╔═╡ 37a3a9c0-4d72-4513-b685-4349f7e8ad20
md"
$\;$
$5040 = \cases{
7 * factorial(6) \cr
7 * (6 * factorial(5)) \cr
7 * (6 * (5 * factorial(4))) \cr
7 * (6 * (5 * (4 * factorial(3)))) \cr
7 * (6 * (5 * (4 * (3 * factorial(2))))) \cr
7 * (6 * (5 * (4 * (3 * (2 * factorial(1)))))) \cr
7 * (6 * (5 * (4 * (3 * (2 * 1)))))) \cr
7 * (6 * (5 * (4 * (3 * 2)))) \cr
7 * (6 * (5 * (4 * 6))) \cr
7 * (6 * (5 * 24)) \cr
7 * (6 * 120) \cr
7 * 720 \cr
5040
}$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
"
# ╔═╡ e469a12c-c66e-4111-ab7a-d1a71281f64a
md"
**Fig. 1.2.1.1**: A *linear* recursive *computation* process with *time* (vertical axis) and *space* (horizontal axis) *complexity* *O(n)* for computing $7!$ (c.f. Fig. 1.3, SICP, 1996)
"
# ╔═╡ 27d1c562-2b1c-41a0-aec9-787834d9d8b4
factorial(7)
# ╔═╡ 1c6a3681-c10f-4ffb-9e45-b5e1601ef8df
md"
###### 3.3 *Tail-Recursive* (Iterative) $factorial$
If the method of the function $factorial$ is coded *tail-recursive* the generated *computational* process may posses *O(1)* *space complexity*. But this will become only true, when the language (like Scheme) has [*tail-call optimization (tco)*](https://en.wikipedia.org/wiki/Tail_call). That means the stack is *not* growing with each new recursive function call. This is *not* true for Julia because Julia lacks *tco*. To obtain *O(1) space complexity* you have replace recursive calls by *iterative loop* constructs like $while$ or $for$.
"
# ╔═╡ 4a17ecb7-9460-48cb-855f-4a49a21bbc9d
fact_iter(product, counter, max_count) =
if >(counter, max_count)
product
else
fact_iter( *(counter, product), +(counter, 1), max_count)
end # fact_iter
# ╔═╡ 5d1ec6ed-ce6a-424e-a2af-47b42cd6eb5b
factorial2(n) = fact_iter(1, 1, n)
# ╔═╡ d746cb2f-2768-4917-b214-583080b0cd1b
factorial2(6)
# ╔═╡ bac2caf9-bf94-4aa0-9ce4-2ea462f4f345
factorial2(6.0)
# ╔═╡ cf1c416b-e4f9-4f24-bf3f-57a28fc67016
md"
$\;$
$5040 = \cases{
factorial(7) \cr
fact\_iter(1, 1, 7) \cr
fact\_iter(1, 2, 7) \cr
fact\_iter(2, 3, 7) \cr
fact\_iter(6, 4, 7) \cr
fact\_iter(24, 5, 7) \cr
fact\_iter(120, 6, 7) \cr
fact\_iter(720, 7, 7) \cr
fact\_iter(5040, 8, 7) \cr
5040
}$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
"
# ╔═╡ c05ac12d-976d-469d-bb79-2f3fdd7341bc
md"
**Fig. 1.2.1.2**: A *tail* recursive *computation* process with *O(n) time complexity* (vertical axis) and *O(1)* or *O(n) space complexity* (horizontal axis) for computing $7!$ (c.f. Fig. 1.4, SICP, 1996) depending on *TCO*
"
# ╔═╡ b484fc01-b9a3-41c0-a844-18efc2c2a800
md"
###### 3.4 *Tail-Recursive* $factorial$ with *Block* Structure
"
# ╔═╡ 68a5453c-9c25-4539-a3f8-2d4948dca25d
function factorial3(n)
#----------------------------------------------------------------
function fact_iter(product, counter, max_count)
if >(counter, max_count)
product
else
fact_iter( *(counter, product), +(counter,1), max_count)
end # if
end # function fact_iter
#----------------------------------------------------------------
fact_iter(1, 1, n)
end # function factorial3
# ╔═╡ b1d0cf2b-dde7-48c9-8cb4-99a37576a85a
factorial3(6)
# ╔═╡ 3f79201d-bd6e-41b2-aa7e-9a3cc6212c0b
factorial3(6.)
# ╔═╡ 9d64350e-6a59-4ebe-b8b8-2c0db70b3987
md"
---
##### 4. Idiomatic *iterative* JULIA ...
... with type 'Signed', *self-defined* abstract type 'FloatOrSigned', 'while', 'for', and *update* operators '+=', '+='
"
# ╔═╡ 6553e3ee-01a8-4883-8b37-0623c8dddb1c
md"
###### 4.1 Linear-Recursive $factorial$ with *Predicate* $isone$
"
# ╔═╡ 82278317-7805-4fd0-9145-1b01a5ccddcd
function factorial4(n)
isone(n) ?
1 :
*(n, factorial( -(n, 1)))
end # function factorial4
# ╔═╡ 72c54616-7701-401d-932e-22d7eedeb297
factorial4(6)
# ╔═╡ 82c84963-2f87-4023-8134-32e36d164d1b
subtypes(Signed) # abstract type Signed
# ╔═╡ 6e6fa605-3d25-426c-98f7-03dabc9eff6f
Integer = Signed # redefinition
# ╔═╡ 440fc1d0-f088-430c-af05-036ccd7986d4
subtypes(Integer) # abstract type Integer: subtype Bool undesired here
# ╔═╡ 8c8fe78d-dddc-4034-87ad-76b2b47cc875
md"
###### 4.2 *Iterative* $factorial$ with $while$ and *Abstract* Type $Integer$
"
# ╔═╡ 03a0d7a7-a12f-45bf-8d24-2c640179166a
#------------------------------------------------------------------
# idiomatic JULIA with 'while'
#------------------------------------------------------------------
function factorial5(n::Integer)::Integer # Float input is blocked!
product = counter = 1
max_count = n
while !(counter > max_count)
product = counter * product
counter = counter + 1
end # while
product
end # function factorial5
#------------------------------------------------------------------
# ╔═╡ 42d11416-24b0-4bd0-bda6-b8c44b3780fc
factorial5(6)
# ╔═╡ da69a9e9-6644-4086-80ab-fc2a0ae20b0b
factorial5(6.) # Float is blocked !
# ╔═╡ 5ed10883-31fb-4f0a-957b-31a746ae0346
md"
###### 4.3 *Iterative* $factorial$ with $while$ and *Abstract* Type $AbstractFloat$
"
# ╔═╡ 83b0a30b-1923-41b7-9b49-396b98bd597d
#-----------------------------------------------------------
# idiomatic JULIA with 'while' and update '*=' and '+='
#-----------------------------------------------------------
function factorial6(n::AbstractFloat)::AbstractFloat
product = counter = 1
max_count = n
while !(counter > max_count)
product *= counter
counter += 1
end # while
product
end # unction factorial6
# ╔═╡ 653e8ea9-37ca-4326-8e93-753e762c99d5
factorial6(6) # Integer is blocked !
# ╔═╡ cc773ec8-28b5-427c-bfe1-a96f9cf72365
factorial6(6.)
# ╔═╡ 9eeea542-4f9d-40eb-a184-4e94ef679797
md"
###### 4.4 *Iterative* $factorial$ with $for$ and *Abstract* Type $FloatOrSigned$
"
# ╔═╡ 8d102915-60f0-417d-b8cf-eff22f2abd11
FloatOrSigned = Union{AbstractFloat, Signed}
# ╔═╡ 0e1023eb-b9b6-42b6-8a4e-6287a735309f
md"
###### 4th self-defined abstract *typed* 'FloatOrSigned' (specialized) method of function 'factorial'
"
# ╔═╡ c0a7751d-6f41-466d-810f-a3886e766956
#-----------------------------------------------------------
# idiomatic JULIA with 'for' and update '*='
#-----------------------------------------------------------
function factorial7(n::FloatOrSigned)::FloatOrSigned
product = 1
for counter = 1:n
product *= counter
end # for
product
end # function factorial7
# ╔═╡ 76619032-3414-42de-b5df-f7299f499413
factorial7(6)
# ╔═╡ 705d89bf-ed5c-4cbb-8d1d-febe7867a785
factorial7(6.)
# ╔═╡ 48735d5e-645a-421b-b65e-37b496e556fb
md"
---
##### 5. Summary
The content of this chapter can be summarized in Fig. 1.2.1.2:
$\begin{array}{c|c|c|c|c|}
\text{} & time & space & time & space \\
\text{} & complexity & complexity & complexity & complexity \\
\text{} & TCO & TCO & no\ TCO & no\ TCO \\
\text{ code } & e.g.Scheme & e.g. Scheme & e.g. Julia & e.g. Julia \\
\hline
\text{ linear recursive } & O(n) & O(n) & O(n) & O(n) \\
\text{ tail recursive } & O(1) & O(1) & O(n) & O(n) \\
\text{ iterative } while, for & -- & -- & O(1) & O(1) \\
\hline
\end{array}$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
$\;$
**Fig. 1.2.1.3**: Relation between *Code, TCO*, and *Complexity*
"
# ╔═╡ df32a460-22c6-4a7a-ac7d-0a14dbf5e6a3
md"
---
##### 6. References
- **Abelson, H., Sussman, G.J. & Sussman, J.**, Structure and Interpretation of Computer Programs, Cambridge, Mass.: MIT Press, (2/e), 1996, 2016, [https://sarabander.github.io/sicp/](https://sarabander.github.io/sicp/), last visit 2022/08/23
- **Sanella, D., Fourman, M., Peng, H. & Wadler, Ph.**; *Introduction to Computation: Haskell, Logic and Automata*; Cham, Switzerland, Springer, 2021
- **Wikipedia**; [*Big O Notation*](https://en.wikipedia.org/wiki/Big_O_notation#Family_of_Bachmann%E2%80%93Landau_notations); last visit 2025/01/06
- **Wikipedia**; [*Mathematical Induction*](https://en.wikipedia.org/wiki/Mathematical_induction); last visit 2025/01/07
- **Wikipedia**; [*tail-call*](https://en.wikipedia.org/wiki/Tail_call); last visit 2025/01/06
"
# ╔═╡ 75bb24f7-0a7c-4165-92c7-aa00d3e8721c
md"
---
##### end of ch. 1.2.1
====================================================================================
This is a **draft** under the [Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)**](https://creativecommons.org/licenses/by-nc-sa/4.0/) license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**
====================================================================================
"
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