Non sequential computing environment and Forth -2-

In part -1- of this series this table was presented to to show some Forth words.
The header part of the table is in bold.

Table 1

	TOS	n1	n2	n3
SWAP	TOS	n2	n1	n3
DROP	TOS - 1		n2	n3
+	TOS - 1		n1 + n2	n3

This table is not (yet) a real Forth table and also can be simplified.
Calculations in this table are 'normal algebra' and not in Forth RPN calculations.
'Normal algebra' n1 + n2 becomes in RPN  n1 n2 + 
As already mentioned in my previous post the change of TOS depends on the amount of bits. The TOS should perhaps should be decremented by 2 or another value depending on the amount of bits in each stack item and the memory width.  It is a system constant depending on the implementation.
In this article series i will use SWDTH (stack width). In the RPN table  TOS - 1 should be written as TOS  SWDTH -

Table 2

	TOS	n1	n2	n3
SWAP	TOS	n2	n1	n3
DROP	TOS SWDTH -		n2	n3
+	TOS SWDTH -		n1 n2 +	n3

Some things in this table are still confusing (however it looks already more Forth).
For example a 'strange historical convention' of n1 n2 n3 where
n1 is the item on top of the stack,  n2 is the item just below it and n3 is two steps below. Another way is to see n1 n2 n3 as memory locations where values calculated are stored.  n1 is TOS @ and n2 is TOS SWDTH - @

Table 3

	TOS	n1	n2	n3
	TOS	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	TOS	n2	n1	n3
DROP	TOS SWDTH -		n2	n3
+	TOS SWDTH -		n1 n2 +	n3

In a real 'physical' implementation of the table it would be needed to include items above the TOS (Top Of the Stack) in case items are added or pushed to the stack. Also it would be strange to represent the first items above the stack using n0 . It would be more systematic if the item on top of stack was referenced as n0 and use n+1 and n-1 for the items just above or below. However in Forth for many years items on the stack where called n1 n2 n3 and breaking with this convention would cause a lot of confusion. Also we do not want tot use complicated difficult to read terms as TOS @ or TOS SWDTH - @  So i decided (temporary) to use p0 and not n1 see tabel below.

Table 4

	TOS	( n0 ? )	n1	n2	n3
	TOS	p1	p0	p-1	p-2
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	TOS		n2	n1	n3
DROP	TOS SWDTH -			n2	n3
+	TOS SWDTH -			n1 n2 +	n3

In the next evolution of the table i added an example of a Forth word that pushes a value to the stack.
It is the Forth word 0 . It is a number implemented in this Forth table where the value 0 is pushed to the stack.
In the table header a new Forth word PA is introduced ( shorthand for ParameterStackAddress ).
The location is described using Forth word  PA to avoid using n1 n2 n3 or p0 p1 p2 notations, 

 : PA SWDTH * TOS @ + ;

Table 5

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
Location	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	TOS		n2	n1	n3
DROP	TOS SWDTH -			n2	n3
+	TOS SWDTH -			n1 n2 +	n3
0	TOS SWDTH +	0	n1	n2	n3

We now have almost everything described in Forth and RPN
To evolve the table further we need to do something with the gaps (empty spaces).

In fact there are several options to handle these gaps.
A) Handle a gap as something that not needs to be changed  ( "Ignore" or "don't change" )

Table 6a

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
Location	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	TOS	Ignore	n2	n1	Ignore
DROP	TOS SWDTH -	Ignore	Ignore	Ignore	Ignore
+	TOS SWDTH -	Ignore	Ignore	n1 n2 +	Ignore
0	TOS SWDTH +	0	Ignore	Ignore	Ignore

B) Fill gap with a "Null" value
"Null" is a special marker and keyword indicating that something has no value (in SQL )
Using  option 2 ( Null) the small table the table would look as below.

Table 6b

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	TOS	Null	n2	n1	n3
DROP	TOS SWDTH -	Null	Null	n2	n3
+	TOS SWDTH -	Null	Null	n1 n2 +	n3
0	TOS SWDTH +	0	n1	n2	n3

Both representations have advantages and disadvantages.
I like option B) using Null as this removes garbage (data that is no longer needed) from the system. Keeping items in a system that are no longer needed can introduce errors and vulnerabilities,

C) A third option could be using Null and, when no changes are needed use some kind of Noop  (no operation, do nothing) indicator.

Table 6c

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	Noop	Null	n2	n1	Noop
DROP	TOS SWDTH -	Null	Null	Noop	Noop
+	TOS SWDTH -	Null	Null	n1 n2 +	Noop
0	TOS SWDTH +	0	Noop	Noop	Noop

If Null and Noop are Forth words a table can be created that describes Forth words in Forth code. The Noop has as disadvantage that just rewriting the old value perhaps is as fast or even faster then  evaluating that something is a Noop and not changing a value.

Table 7

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	Noop	Null	-1 PA	0 PA	Noop
DROP	TOS SWDTH -	Null	Null	Noop	Noop
+	TOS SWDTH -	Null	Null	0 PA -1 PA +	Noop
0	TOS SWDTH +	0	Noop	Noop	Noop

Extending the table with more Forth words gives the next table (Table 8)

Table 8

	TOS	( n0 ? )	n1	n2	n3
		p1	p0	p-1	p-2
	TOS	1 PA	0 PA	-1 PA	-2 PA
	TOS	TOS SWDTH + @	TOS @	TOS SWDTH - @	TOS SWDTH 2* - @
SWAP	Noop	Null	-1 PA	0 PA	Noop
DROP	TOS SWDTH -	Null	Null	Noop	Noop
+	TOS SWDTH -	Null	Null	0 PA -1 PA +	Noop
0	TOS SWDTH +	0	Noop	Noop	Noop
1	TOS SWDTH +	1	Noop	Noop	Noop
2	TOS SWDTH +	2	Noop	Noop	Noop
3	TOS SWDTH +	3	Noop	Noop	Noop
4	TOS SWDTH +	4	Noop	Noop	Noop
-1	TOS SWDTH +	-1	Noop	Noop	Noop
-2	TOS SWDTH +	-2	Noop	Noop	Noop
Noop	Noop	Noop	Noop	Noop	Noop
-	TOS SWDTH -	Null	Null	0 PA -1 PA -	Noop
*	TOS SWDTH -	Null	Null	0 PA -1 PA *	Noop
/	TOS SWDTH -	Null	Null	0 PA -1 PA /	Noop
=	TOS SWDTH -	Null	Null	0 PA -1 PA =	Noop
@	Noop	Null	@	Noop	Noop
!	TOS SWDTH -	Null	Null	Null	Noop
2*	Noop	Null	2 *	Noop	Noop
2*	Noop	Null	2*	Noop	Noop
TOS	TOS SWDTH +	TOS	Noop	Noop	Noop
TOS	TOS SWDTH +	"Value"	Noop	Noop	Noop
SWDTH	TOS SWDTH +	"Value"	Noop	Noop	Noop
SWDTH+	Noop	Null	Noop	Noop	Noop
SWDTH-	Noop	Null	Noop	Noop	Noop

In the last part of table 8 some issues can be seen that need (and can) be solved.  Therefor i did write it in Italics and some words i described in multiple ways.

! Can't be full described in this table.The result of ! will be that 2 items are removed from the stack. An address and a value. The value will be stored at the address. However it is not easy to extend the lookup table so it will include all addresses in memory. ( @ can be described in the table. However there must be some kind of low level implementation!)

2* multiplies the value that is on top of stack by two. It can be described as just doing 2 *
Written in Forth code : 2* 2 * ;  
However in some Forth systems this is implemented in low level code as it can be executed fast,  This is just one example, however in general this creates a dilemma when trying to make a table like this. Many Forth words can easy be defined as itself, However that does not explain what these Forth words do. Also sometimes these words can be implemented in another way using other Forth words or should  call low level code.  

TOS ( Top Of Stack) is a variable. It it can be put in the table just calling some implementation of itself. However an easy implementation is pushing 'the value of the address' on the stack.

 SWDTH is a (system) constant, as described in the beginning of this article. A constant can be described in this table as it self, Or just put the value on the stack comparable to TOS

Some sequences of operations appear very often in this table. Examples are
TOS SWDTH + and TOS SWDTH + and also my new Forth word PA
  : PA SWDTH * TOS @ + ;
This indicates that speeding up these sequences can speed up the Forth system. 

We have seen that we can put the descriptions of (several) Forth words in a table.
Can that help us in a new Forth implementation and what about a 'Non sequential computing environment'? Yes, if we can describe Forth words in a table we can describe Forth in a table. This not only makes Forth portable, it also makes it possible we can create Forth in a table environment, e.g. a relational database.