As explained in the ADP for MCFL page, the main change compared to ADP for CFL is the addition of rewriting functions.

## This is how a nice syntax could look like

S = nil  << empty
| left << B S
| pair << P S S       >> \ (p1,p2) s1 s2 -> p1 s1 p2 s2
| knot << K K S S S S >> \ (k11,k12) (k21,k22) s1 s2 s3 s4 -> k11 s1 k21 s2 k12 s3 k22 s4
... h

B = base << 'a' | base << 'u' | base << 'c' | base << 'g'

P = bpair << ('a','u')
| bpair << ('u','a')
| [...]

K = knot1 << P K >> \ (p1,p2) (k1,k2) -> (k1 p1, p2 k2)
| knot2 << P


Of course, rewriting functions should also be assignable to separate identifiers to allow reuse:

rewriteKnot1 (p1,p2) (k1,k2) = (k1 p1, p2 k2)
K = knot1 << P K >> rewriteKnot1
| knot2 << P


## This is how it actually looks (at the moment)

rewritePair, rewriteKnot :: Dim1

s = tabulated1 $yieldSize1 (0,Nothing)$
nil  <<< empty         >>> id |||
left <<< b ~~~ s       >>> id |||
pair <<< p ~~~ s ~~~ s >>> (\ [p1,p2,s1,s2] -> [p1,s1,p2,s2]) |||
knot <<< k ~~~ k ~~~ s ~~~ s ~~~ s ~~~ s >>> (\ [k11,k12,k21,k22,s1,s2,s3,s4] -> [k11,s1,k21,s2,k12,s3,k22,s4])
... h

b = tabulated1 $base <<< 'a' >>> id ||| base <<< 'u' >>> id ||| base <<< 'c' >>> id ||| base <<< 'g' >>> id p = tabulated2$
bpair <<< ('a','u') >>> id2 |||
bpair <<< ('u','a') >>> id2 |||
[...]

rewriteKnot1 :: Dim2
rewriteKnot1 [p1,p2,k1,k2] = ([k1,p1],[p2,k2])

k = tabulated2 $yieldSize2 (1,Nothing) (1,Nothing)$
knot1 <<< p ~~~ k >>> rewriteKnot1 |||
knot2 <<< p       >>> id2


## Differences explained

### S vs s

As the grammars use parser combinators, their non-terminals are functions. In Haskell, variable identifiers (like a function identifier) must start with a lower-case letter while constructor identifiers (like a data type) must start with an upper-case letter. (Haskell 98 Lexical Structure)

### tabulated1 and tabulated2

ADP compilers like GAP-C can do an automatic table design analysis and then decide which non-terminals should be tabulated. As adp-multi doesn't do any static analysis (except limited yield size analysis, see below) the programmer explicitly has to state which tables should be tabulated. The numbers in tabulated1 and tabulated2 specify the table dimensions which means that tabulated1 will use a two dimensional table and tabulated2 will use a four dimensional table.

Note: At the moment, choosing the wrong version of tabulated* doesn't yield to compilation but runtime errors. This typing problem is explained further below and should be solved in future versions.

### | vs ||| and << vs <<<

In Haskell, | is reserved as a keyword, and || is defined as a function in Prelude which could theoretically be used (by hiding and redefining it, causing more trouble). Although ||| is also defined in Control.Arrow, the conflicts are limited to a minimum which is why it is used.

Using << would be possible but for reasons of symmetry to ||| and >>>, it is defined as <<<.

### yieldSize1 and yieldSize2 (or: How does yield size analysis work?)

The original Haskell-ADP implementation has several combinators to connect two non-terminals (~~~,-~~,~~-,+~~,~~+,+~+,...). Those different variants are necessary to tell the top-down parsing mechanism what the minimum yield sizes are -- otherwise the subword ranges would never get smaller in a recursion, leading to endless recursion.

In GAP-C this isn't necessary anymore because the compiler does a yield size analysis and then knows the minimum and possibly maximum yield sizes of each nonterminal.

In adp-multi, a restricted yield size analysis was built in. Restricted means that it doesn't handle cycles because this would require a full blown abstract syntax tree analysis of the grammar (as in GAP-C). Therefore the grammar writer has to tell the parser where the cycles are and then explicitly cut them off by using yieldSize1 and yieldSize2, respectively. Cutting off means that the minimum and maximum yield size of a nonterminal is specified manually, therefore skipping the yield size analysis.

### >> vs >>>

For symmetry reasons, >>> was chosen for rewriting function.

### >> optional vs required

In original Haskell-ADP, the combinators for connecting two nonterminals were themselves responsible for creating the right subword indices. In adp-multi, this isn't possible anymore due to the rewriting functions. Therefore, the operator >>> is responsible for creating all possible subword indices for a complete rule. In cases where no rewriting is required, the >>> must still be applied with the identity function so that the subword indices are generated. It is not yet clear if this could be done implicitly by using Haskell's type classes.

### \ (p1,p2) s1 s2 vs \ [p1,p2,s1,s2]

Having a rule like val <<< p ~~~ p >>> rewrite and making it typesafe means that the two nonterminal parsers have to be applied one after another both to val and rewrite. This seems rather complicated, considering that each nonterminal parser can have a different dimension. I'm sure that this is solvable with some advanced Haskell extensions like Type Families, GADTs, etc.

For now, to produce a working prototype, the type safety was loosened a bit and lists are used in all such cases. This means that related errors will only be detected at runtime.

If someone has hints on that, I would be very thankful!

## Why can't terminal symbols be included in rewriting functions?

The formalism of MCFGs allows to write rewriting functions like the following:

$$K \rightarrow g[K] \mid (a, b)\\ g[(x_1,x_2)] = (x_1 a, b x_2)$$

Another (less usual) way is:

$$K \rightarrow g[K,a,b] \mid (a, b)\\ g[(x_1,x_2),x_3,x_4] = (x_1 x_3, x_4 x_2)$$

or:

$$K \rightarrow g[K,(a,b)] \mid (a, b)\\ g[(x_1,x_2),(x_3,x_4)] = (x_1 x_3, x_4 x_2)$$

Only the last two ways can be used in adp-multi:

k = val <<< 'a' ~~~ 'b' ~~~ k >>> g
g [x3,x4,x1,x2] = ([x1,x3],[x4,x2])


and

k = val <<< ('a','b') ~~~ k >>> g
g [x3,x4,x1,x2] = ([x1,x3],[x4,x2])


This little restriction made the implementation a lot easier and shouldn't cause any real inconvenience. It might even help in more quickly recognizing all parts of a rule without looking at the rewriting functions.