Modal is a matrioshka language based on pattern-matching to rewrite trees

XXIIVV — modal

Modal In A Postcard

Modal is a matrioshka language based on pattern-matching to rewrite trees.

Modal programs are represented as a series of rules, formatted as tokens delimited by brackets and parentheses, applied to a given tree which gets continually modified until no rules match any given part of the tree.

modal(adj.):
of, or relating to structure as opposed to substance.

Elements of modal are:

Rules: <> left right
Registers: ?x, ?y, ?z
Trees: word, (a tuple), a (cons (list))

The documentation below displays the examples as a series of rules, followed by the rewriting steps in the following format:

<> A rule

.. The input program
04 The result of applying rule #4
-1 The result of applying a lambda

Rules

To define a new rule, start with <>, followed by a left and a right statement, which is either a word, or a tree. The program evaluation starts at the first character of the string and walks through to the end trying to match a transformation rule from that location:

<> hello (good bye)  This is a rule

.. hello world       This is program data
00 good bye world    This is the result

Registers

Registers are single-character identifiers bound to an address in a pattern used in rewriting:

<> (copy ?a) (?a ?a)

.. copy cat
00 cat cat

When a register is used in a pattern, and when we try to match a given tree with a pattern, each register is bound to a corresponding an address and referenced in either side of a rule:

<> (swap ?x ?y) (?y ?x)

.. (swap fox rat)
00 (rat fox)

When a register appears more than once in a rule, each instance is bound to the first address:

<> (?x ?x ?x) triplet

.. (fox fox fox)
00 (triplet)

Trees

Trees can be found in rules and program data, they include words, registers and nested trees. Rules can match specific trees and rewrite their content in a new sequence.

<> (rotate ?x (?y) ?z) (?y (?z) ?x)

.. rotate foo (bar) baz
00 bar (baz) foo

An efficient way to represent an array is to store information in nested lists, it allows for rules to target specific segments of the list, similarly to Lisp's car and cdr primitives. To print each element of such a structure, we can use the following recursive rules:

<> (putrec (?: ?x)) (putrec ?:?x)
<> ((putrec (?:))) (?:)

.. (putrec (a (b (c (d (e))))))
00 (putrec (b (c (d (e)))))
00 (putrec (c (d (e))))
00 (putrec (d (e)))
00 (putrec (e))
01 

> abcde

Special Registers

Special registers are registers that do more than simply store a reference, they allow implementations to choose which special behavior is needed by the host platform, without impacting the core of the language:

Substrings
Explode token	?(?* ?*) abc	a (b (c ()))
Explode tuple	?(?* ?*) (abc def ghi)	abc (def (ghi ()))
Unpack	?(?. ?.) (abc def)	abc def
Join	?(?^ ?^) (abc def ghi)	abcdefghi
IO
Read	?~	Read from devices
Send	?:	Send to devices

A lambda is created by using the ?(body) special register. Rules created that way exist only for the length of one rewrite and must match what is found immediately after:

.. ?((?x ?y) (?y ?x)) foo bar
-1 bar foo

Explode a token or tuple, into a nested list with the ?* special register, notice how the following program makes use the List type to ensure a specific evaluation order:

<> (reverse List () ?^) (?^)
<> (reverse (?*)) (reverse List (?*) ())
<> (reverse List (?x ?y) ?z) (reverse List ?y (?x ?z))

.. (reverse (modal))
01 (reverse List (m (o (d (a (l ()))))) ())
02 (reverse List (o (d (a (l ())))) (m ()))
02 (reverse List (d (a (l ()))) (o (m ())))
02 (reverse List (a (l ())) (d (o (m ()))))
02 (reverse List (l ()) (a (d (o (m ())))))
02 (reverse List () (l (a (d (o (m ()))))))
00 (ladom)

Sending a message to a device is done with the ?: special register, it sends a word or a tree to be handled by a device:

<> (print ?:) (?:)

.. print (hello world\n)

hello world

Similarly, reading an incoming message from a device is done with the ?~ special register:

<> (?: print) (?:)
<> (READ ?~) ((You said: ?~ \n) print)

.. (READ stdin)

You said:

Type Systems

Understanding how to use types to guard rules for specific evaluation order is important to become proficient with Modal. Creating a type system is merely a matter of creating stricter rules expecting a specific grammar.

<> (join ?^) (?^)
<> (join-strings (String ?x) (String ?y)) (join (?x ?y))

.. join-strings (String foo) (String bar)
01 join (foo bar)
00 foobar

Notice in the example above, how join-strings expects to match two String typed words. Without typed inputs, the rule is not matched.

.. join-string (bar baz)

Logic

Let us build a logic system, starting by comparing two registers:

<> (eq ?x ?x) (#t)
<> (eq ?x ?y) (#f)

.. (eq fox bat)
01 (#f)

We can implement the truth tables by defining each case:

<> (and #t #t) #t <> (or #t #t) #t 
<> (and #t #f) #f <> (or #t #f) #t
<> (and #f #t) #f <> (or #f #t) #t 
<> (and #f #f) #f <> (or #f #f) #f
<> (not #t) #f    <> (not #f) #t

.. (or #f #t)
08 (#t)

Building on the comparison rule above, we can write conditionals with a ternary statement:

<> (ife #t ?t ?f) (?t)
<> (ife #f ?t ?f) (?f)
<> (print ?:) (?:)

.. ife #f (print True!) (print False!)
13 (print False!)
14 ()

Arithmetic

The language does not accommodate for any specific numerical system, but allows for the notion of numbers to be implemented with Peano Numerals:

<> (add (s ?x) (s ?y)) (s (add ?x (s ?y)))
<> (add (s ?x) (0)) (s ?x)
<> (add (0) (s ?y)) (s ?y)
<> (add (0) (0)) (0)
<> (sub (s ?x) (s ?y)) (sub ?x ?y)
<> (sub (s ?x) (0)) (s ?x)
<> (sub (0) (s ?y)) (s ?y)
<> (sub (0) (0)) (0)
<> (mul (s ?x) (s ?y)) (add (s ?x) (mul (s ?x) (sub (s ?y) (s (0)))))
<> (mul (s ?x) (s (0))) (s ?x)
<> (mul (s (0)) (s ?y)) (s ?y)
<> (mul (s ?x) (0)) (0)
<> (mul (0) (s ?x)) (0)

To convert from prefix notation to infix:

<> (?x + ?y) (add ?x ?y)
<> (?x - ?y) (sub ?x ?y)
<> (?x * ?y) (mul ?x ?y)

Altogether, we have enough parts to implement factorial:

<> (factorial (s (0))) ((s (0)))
<> (factorial (s ?x)) (((s ?x) * factorial ((s ?x) - (s (0)))))

factorial (s (s (s (s (s (0))))))

Binary

Prefix rules to increment a binary number:

<> (inc (0 ?x)) ((1 ?x))
<> (inc (1 ?x)) ((0 inc ?x))
<> (inc ()) ((1 ()))

?(?-) (Count to 0x7f)

<> (> increc (1 (1 (1 (1 ()))))) (done.)
<> (> increc ?i) (> (inc ?i wait) increc)
<> (> (?i wait) increc) (> increc ?i)

> increc ()

Mimics

We can use rules to define entire languages, Modal enforces no specific notation, for example, we could easily make a combinatory logic playground:

<> (M ?x) (?x ?x)
<> (KI ?x ?y) (?y)
<> (T ?x ?y) (?y ?y)
<> (W ?x ?y) (?x ?y ?y)
<> (K ?x ?y) (?x)
<> (C ?x ?y ?z) (?x ?z ?y)
<> (B ?x ?y ?z) (?x (?y ?z))
<> (I ?x) (?x)
<> (S ?x ?y ?z) (?x ?z (?y ?z))

.. C KI x y z
05 KI y x z
01 x z

Any choice made in regard to syntax is completely arbitrary. To demonstrate, the following code defines a concatenative syntax:

<> (?x dup) (?x ?x)
<> (?x ?y swap) (?y ?x)
<> (?x pop) ()

.. (1 2 3) (4 5 6) swap pop dup
01 (4 5 6) (1 2 3) pop dup
02 (4 5 6) dup
00 (4 5 6) (4 5 6)

Homoiconicity is a property of some programming languages that treats code as data, when the internal and external representation of a program is the same. Modal is homoiconic, as any string is a potential program and new rules can be composed directly during the evaluation. For instance, here is a rule to define new rules with an infix syntax:

<> ((?x -> ?y)) (<> ?x ?y)
(a -> apple)
(b -> banana)
((apple banana) -> (fruit-salad))

.. a b
01 apple b
02 apple banana
03 fruit-salad

Implementation

The Modal runtime can be implemented in about 200 lines of ANSI C.

cc modal.c -o modal view raw

view sources, ANSI C.
discord channel, in the concatenative server.
Original creation of wryl, many of the code above is their own work and merely made available here as to give this fantastic system a home on the internet.