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Multi-Directional Pattern Notation

Final - Sep 6 1999


MDPN is an extension to EBNF, which attributes it for the purposes of scanning and parsing input files which assume a non-unidirectional form. A familiarity with EBNF is assumed for the remainder of this document.

MDPN was developed by Chris Pressey in late 1998, built on an earlier, less successful attempt at a "2D EBNF" devised to fill a void that the mainstream literature on parsing seemed to rarely if ever approach, with much help provided by John Colagioia throughout 1998.

MDPN has possible uses in the construction of parsers and subsequently compilers for multi-directional and multi-dimensional languages such as Orthogonal, Befunge, Wierd, Blank, Plankalk├╝l, and even less contrived notations like structured Flowchart and Object models of systems.

As the name indicates, MDPN provides a notation for describing multidimensional patterns by extending the concept of linear scanning and matching with geometric attributes in a given number of dimensions.

Preconditions for Multidirectional Parsing

The multidirectional parsing that MDPN concerns itself with assumes that any portion of the input file is accessable at any time. Concepts such as LL(1) are fairly meaningless in a non-unidirectional parsing system of this sort. The unidirectional input devices such as paper tape and punch cards that were the concern of original parsing methods have been superceded by modern devices such as hard disk drives and ample, cheap RAM.

In addition, MDPN is limited to an orthogonal representation of the input file, and this document is generally less concerned about forms of four or higher dimensions, to reduce unnecessary complexity.

Notation from EBNF

Syntax is drawn from EBNF. It is slightly modified, but should not surprise anyone who is familiar with EBNF.

A freely-chosen unadorned ('bareword') alphabetic multicharacter identifier indicates the name of a nonterminal (named pattern) in the grammar. e.g. foo. (Single characters have special meaning as operators.) Case matters: foo is not the same name as Foo or FOO.

Double quotes begin and end literal terminals (symbols.) e.g. "bar".

A double-colon-equals-sign (::=) describes a production (pattern match) by associating a single nonterminal on the left with a pattern on the right, terminated with a period. e.g. foo ::= "bar".

A pattern is a series of terminals, nonterminals, operators, and parenthetics.

The | operator denotes alternatives. e.g. "foo" | "bar"

The ( and ) parentheses denote precedence and grouping.

The [ and ] brackets denote that the contents may be omitted, that is, they may occur zero or one times. e.g. "bar" ["baz"]

The { and } braces denote that the contents may be omitted or may be repeated any number of times. e.g. "bar" {baz "quuz"}

Deviations from EBNF

The input file is spatially related to a coordinate system and it is useful to think of the input mapped to an orthogonally distributed (Cartesian) form with no arbitrary limit imposed on its size, hereinafter referred to as scan-space.

The input file is mapped to scan-space. The first printable character in the input file always maps to the origin of scan-space regardless of the number of dimensions. The origin is enumerated with coordinates (0) in one dimension, (0,0) in two dimensions, (0,0,0) in three dimensions, etc.

Scan-space follows the 'computer storage' co-ordinate system so that x coordinates increase to the 'east' (rightwards), y coordinates increase to the 'south' (downwards), and z coordinates increase on each successive 'page'.

Successive characters in the input file indicate successive coordinate (x) values in scan-space. For two and three dimensions, end-of-line markers are assumed to indicate "reset the x dimension and increment the y dimension", and end-of-page markers indicate "reset the y dimension and increment the z dimension", thus following the commonplace mapping of computer text printouts.

Whitespace in the input file are not ignored. The terminal " ", however, will match any whitespace (including tabs, which are not expanded.) The pattern {" "} may be used to indicate any number of whitespaces; " " {" "} may be used to indicate one or more whitespaces. Areas of scan-space beyond the bounds of the input file are considered to be filled with whitespaces.

Therefore, "hello" as a terminal is exactly the same as "h" "e" "l" "l" "o" as an pattern of terminals.

A } closing brace can be followed by a ^ (constraint) operator, which is followed by an expression in parentheses.

This expression is actually in a subnotation which supports a very simple form of algebra. The expression (built with terms connected by infix +-*/% operators with their C language meanings) can either reduce to

  • a constant value, as in {"X"} ^ (5), which would match five X terminals in a line; or
  • an unknown value, which can involve any single lowercase letters, which indicate variables local to the production, as in {"+"}^(x) {"-"}^(x*2), which would match only twice as many minus signs as plus signs.

Complex algebraic expressions in constraints can and probably should be avoided when constructing a MDPN grammar for a real (non-contrived) compiler. MDPN-based compiler-compilers aren't expected to support more than one or two unknowns per expression, for example. There is no such restriction, of course, when using MDPN as a guide for hand-coding a multidimensional parser, or otherwise using it as a more sophisticated pattern-matching tool.

The Scan Pointer

It is useful to imagine a scan pointer (SP, not to be confused with a stack pointer, which is not the concern of this document) which is analogous to the current token in a single-dimensional parser, but exists in MDPN as a free spatial relationship to the input file, and thus also has associated geometric attributes such as direction.

The SP's location is advanced through scan-space by its heading as terminals in the productions are successfully matched with symbols in the input buffer.

The following geometric attribution operators modify the properties of the SP. Note that injudicious use of any of these operators can result in an infinite loop during scanning. There is no built-in contingency measure to escape from an infinite parsing loop in MDPN (but see exclusivity, below, for a possible way to overcome this.)

t is the relative translation operator. It is followed by a vector, in parentheses, which is added to the location of the SP. This does not change its heading.

For example, t (0,-1) moves the SP one symbol above the current symbol (the symbol which was about to be matched.)

As a more realistic example of how this works, consider that the pattern "." t(-1,1) "!" t(0,-1) will match a period with an exclamation point directly below it, like:


r is the relative rotation operator. It is followed by an axis identifier (optional: see below) and an orthogonal angle (an angle a such that |a| mod 90 degrees = 0) assumed to be measured in degrees, both in parentheses. The angle is added to the SP's heading. Negative angle arguments are allowed.

Described in two dimensions, the (default) heading 0 denotes 'east,' that is, parsing character by character in a rightward direction, where the SP's x axis coordinate increases and all other axes coordinates stay the same. Increasing angles ascend counterclockwise (90 = 'north', 180 = 'west', 270 = 'south'.)

For example, ">" r(-90) "+^" would match


The axis identifier indicates which axis this rotation occurs around. If the axis identifier is omitted, the z axis is to be assumed, since this is certainly the most common axis to rotate about, in two dimensions.

If the axis identifier is present, it may be a single letter in the set xyz (these unsurprisingly indicate the x, y, and z dimensions respectively), or it may be a non-negative integer value, where 0 corresponds to the x dimension, 1 corresponds to the y dimension, etc. (Implementation note: in more than two dimensions, the SP's heading property should probably be broken up internally into theta, rho, &c components as appropriate.)

For example, r(z,180) rotates the SP's heading 180 degrees about the z (dimension #2) axis, as does r(2,180) or even just r(180).

< and > are the push and pop state-stack operators, respectively. Alternately, they can be viewed as lookahead-assertion parenthetics, since the stack is generally assumed to be local to the production. (Compiler-compilers should probably notify the user, but not necessarily panic, if they find unbalanced <>'s.)

All properties of the SP (including location and heading, and scale factor if supported) are pushed as a group onto the stack during < and popped as a group off the stack during >.

Advanced SP Features

These features are not absolutely necessary for most non-contrived multi-directional grammars. MDPN compiler-compilers are not expected to support them.

T is the absolute translation operator. It is followed by a vector which is assigned to the location of the SP. e.g. T (0,0) will 'home' the scan.

R is the absolute rotation operator. It is followed by an optional axis identifier, and an orthogonal angle assumed to be measured in degrees. The SP's heading is set to this angle. e.g. R(270) sets the SP scanning line after line down the input text, downwards. See the r operator, above, for how the axis identifier functions.

S is the absolute scale operator. It is followed by an orthogonal scaling factor (a scalar s such that s = int(s) and s >= 1). The SP's scale factor is set to this value. The finest possible scale, 1, indicates a 1:1 map with the input file; for each one input symbol matched, the SP advances one symbol in its path. When the scale factor is two, then for each one input symbol matched, the SP advances two symbols, skipping over an interim symbol. Etc.

s is the relative scale operator. It is followed by a scalar integer which is added to the SP's scaling factor (so long as it does not cause the scaling factor to be zero or negative.)

Scale operators may also take an optional axis identifier (as in S(y,2)), but when the axis identifier is omitted, all axes are assumed (non-distortional scaling).

!> is a state-assertion alternative to >, for the purpose of determining that the SP successfully and completely reverted to a previous state that was pushed onto the stack ('came full circle'). This operator is something of a luxury; a grammar which uses constraints correctly should never need it, but it can come in handy.

Other Advanced Features: Exclusivity

Lastly, in the specification of a production, the exclusivity applying to that production can be given between a hyphen following the name of the nonterminal, and the ::= operator.

Exclusivity is a list of productions, named by their nonterminals, and comes into play at any particular instance of the production (i.e. when the production successfully matches specific symbols at specific points in scan-space during a parse, called the domain.) The exclusivity describes how the domain of each instance is protected from being the domain of any further instances. The domain of any subsequent instances of any productions listed in the exclusivity is restricted from sharing points in scan-space with the established domain.

Exclusivity is a measure to prevent so-called crossword grammars - that is, where instances of productions can overlap and share common symbols - if desired. Internally it's generally considered a list of 'used-by-this-production' references associated with each point in scan-space. An example of the syntax to specify exclusivity is bar - bar quuz ::= foo {"a"} baz. Note that the domain of an instance of bar is the sum of the domains foo, baz and the chain of "a" terminals, and that neither a subsequent instance of quuz nor bar again can overlap it.

Examples of MDPN-described Grammars

Example 1. A grammar for describing boxes.

The task of writing a translator to recognize a two-dimensional construct such as a box can easily be realized using a tool such as MDPN.

An input file might contain a of box drawn in ASCII characters, such as

|      |
|      |

Let's also say that boxes have a minimum height of four (they must contain at least two rows), but no minimum width. Also, edge characters must match up with which edge they are on. So, the following forms are both illegal inputs:


|   |
|   |

The MDPN production used to describe this box might be

  Box ::= "+" {"-"}^(w) r(-90) "+" "||" {"|"}^(h) r(-90)
          "+" {"-"}^(w) r(-90) "+" "||" {"|"}^(h) r(-90).

Example 2. A simplified grammar for Plankalk├╝l's assignments.

An input file might contain an ASCII approximation of something Zuse might have jotted down on paper:

 |Z   + Z   => Z
V|1     2      3
S|1.n   1.n    1.n

Simplified MDPN productions used to describe this might be

Staff     ::= Spaces "|" TempVar AddOp TempVar Assign TempVar.
TempVar   ::= "Z" t(-1,1) Index t(-1,1) Structure t(0,-2) Spaces.
Index     ::= <Digit {Digit}>.
Digit     ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9".
Structure ::= <Digit ".n">.
AddOp     ::= ("+" | "-") Spaces.
Assign    ::= "=>" Spaces.
Spaces    ::= {" "}.