An example of a symch

Hesperus is Venus's traditional name as an evening star, and Phosphorus (aka Lucifer) is its traditional name as a morning star. For the grammar,

    :start ::= planet
    planet ::= hesperus
    planet ::= phosphorus
    hesperus ::= venus
    phosphorus ::= venus
    venus ~ 'venus'

and the input string '"venus"', the forest would look like

    Symbol #0 planet has 2 symches
      Symch #0.0
      GL2 Rule 0: planet ::= hesperus
        GL3 Rule 2: hesperus ::= venus
          GL4 Symbol venus: "venus"
      Symch #0.1
      GL2 Rule 1: planet ::= phosphorus
        GL5 Rule 3: phosphorus ::= venus
          GL6 Symbol venus: "venus"

Notice the tags of the form ""GLn"", where n is an integer. These identify the glade. Glades will be described in detail below.

The rules allow the string '"venus"' to be parsed as either one of two planets: '"hesperus"' or '"phosphorus"', depending on whether rule 0 or rule 1 is used. The choice, at glade 2, between rules 0 and 1, is a symch.

An example of a factoring

For the grammar,

    :start ::= top
    top ::= b b
    b ::= a a
    b ::= a
    a ~ 'a'

and the input '"aaa"', a successful parse will always have two "b"'s. Of these two "b"'s one will always be short, deriving a string of length 1: '"a"'. The other will always be long, deriving a string of length 2: '"aa"'. But they can be in either order, which means that the two "b"'s can divide up the input stream in two different ways: long string first; or short string first.

These two different ways of dividing the input stream using the rule

    top ::= b b

are called a factoring. Here's Marpa's dump of the forest:

    GL2 Rule 0: top ::= b b
      Factoring #0
        GL3 Rule 2: b ::= a
          GL4 Symbol a: "a"
        GL5 Rule 1: b ::= a a
          GL6 Symbol a: "a"
          GL7 Symbol a: "a"
      Factoring #1
        GL8 Rule 1: b ::= a a
          GL9 Symbol a: "a"
          GL10 Symbol a: "a"
        GL11 Rule 2: b ::= a
          GL12 Symbol a: "a"

Glades

Each glade node represents an instance of a symbol in one of the possible parse trees. This means that each glade has a symbol (called the "glade symbol"), and an "input span". An input span is an input start location, and a length in characters. Because it has a start location and a length, a span also specifies an end location in the input.

Symches

Every glade contains one or more symches. If a glade has only one symch, that symch is said to be trivial. A symch is either a token symch or a rule symch. For a token symch, the glade symbol is the token symbol. For a rule symch, the glade symbol is the LHS of the rule.

At most one of the symches in a glade can be a token symch. There can, however, be many rule symches in a glade -- one for every rule with the glade symbol on its LHS.

Factorings

Each rule symch contains one or more factorings. A factoring is a way of dividing up the input span of the glade among its RHS symbols, which in this context are called factors. If a rule symch has only one factoring, that factoring is said to be trivial. A token symch contains no factorings, which means that token symches are the terminals of an ASF.

Because the number of factorings can get out of hand, factorings may be omitted. A symch which omits factorings is said to be truncated. By default, every symch is truncated down to its first 42 factorings.

Factors

Every factoring has one or more factors. Each "factor" corresponds to a symbol instance on the RHS of the rule. Each such RHS factor is also a downglade, one which contains its own symches.

Memoization

When traversing a forest, you should take steps to avoid traversing the same glades twice. You can do this by memoizing the result of each glade, perhaps using its glade ID to index an array. When a glade is visited, the array can be checked to see if its result has been memoized. If so, the memoized result should be used.

This memoization eliminates the need to revisit the downglades of an already visited glade. It does not eliminate multiple visits to a glade, but it does eliminate retraversal of the glades downhill from it. In practice, the improvement in speed can be stunning. It will often be the difference between an program which is unuseably slow even for very small inputs, and one which is extremely fast even for large inputs.

Repeated subtraversals happen when two glades share the same downglades, something that occurs frequently in ASF's. Additionally, some day the SLIF may allow cycles. Memoization will prevent a cycle form causing an infinite loop.

The example in this POD includes a memoization scheme which is very simple, but adequate for most purposes. The main logic of its memoization is shown here.

        my ( $asf, $glade, $seen ) = @_;
        return bless ["Glade $glade revisited"], 'My_Revisit'
            if $seen->[$glade];
        $seen->[$glade] = 1;

Putting memoization in one of the very first drafts of your code will save you time and trouble.

peak()

    my $peak = $asf->peak();

Returns the glade ID of the peak. This may be zero. All failures are thrown as exceptions.

glade_literal()

        my $literal = $asf->glade_literal($glade);

Returns the literal substring of the input associated with the glade. Every glade is associated with a span -- a start location in the input, and a length. On failure, throws an exception.

The literal is determined by the range. This works as expected if your application reads the input characters one-by-one in order. (We will call applications which read in this fashion, monotonic.) Most applications are monotonic, and yours is, unless you've taken special pains to make it otherwise. Computation of literal substrings for non-monotonic applications is addressed in "Literals and G1 spans" in Marpa::R2::Scanless::R.

glade_span()

    my ( $glade_start, $glade_length ) = $asf->glade_span($glade_id);

Returns the span of the input associated with the glade. Every glade is associated with a span -- a start location in the input, and a length. On failure, throws an exception.

The span will be as expected if your application reads the input characters one-by-one in order. (We will call applications which read in this fashion, monotonic.) Most applications are monotonic, and yours is, unless you've taken special pains to make it otherwise. Computation of literal substrings for non-monotonic applications is addressed in "Literals and G1 spans" in Marpa::R2::Scanless::R.

glade_symch_count()

    my $symch_count = $asf->glade_symch_count($glade);

Requires a glade ID as its only argument. Returns the number of symches contained in the glade specified by the argument. On failure, throws an exception.

glade_symbol_id()

    my $symbol_id    = $asf->glade_symbol_id($glade);
    my $display_form = $grammar->symbol_display_form($symbol_id);

Requires a glade ID as its only argument. Returns the symbol ID of the "glade symbol" for the glade specified by the argument. On failure, throws an exception.

symch_rule_id()

    my $rule_id = $asf->symch_rule_id( $glade, $symch_ix );

Requires two arguments: a glade ID and a zero-based symch index. These specify a symch. If the symch specified is a rule symch, returns the rule ID. If it is a token symch, returns -1.

Returns a Perl undef, if the glade exists, but the symch index is too high. On other failure, throws an exception.

symch_is_truncated()

[ To be written. ]

symch_factoring_count()

    my $factoring_count =
        $asf->symch_factoring_count( $glade, $symch_ix );

Requires two arguments: a glade ID and a zero-based symch index. These specify a symch. Returns the count of factorings if the specified symch is a rule symch. This count will always be one or greater. Returns zero if the specified symch is a token symch.

Returns a Perl undef, if the glade exists, but the symch index is too high. On other failure, throws an exception.

factoring_downglades()

    my $downglades =
        $asf->factoring_downglades( $glade, $symch_ix,
        $factoring_ix );

Requires three arguments: a glade ID, the zero-based index of a symch and the zero-based index of a factoring. These specify a factoring. On success, returns a reference to an array. The array contains the glade IDs of the the downglades in the factoring specified.

Returns a Perl undef, if the glade and symch exist, but the factoring index is too high. On other failure, throws an exception. In particular, exceptions are thrown if the symch is for a token; and if the glade exists, but the symch index is too high.

ambiguities()

    my $ambiguities = Marpa::R2::Internal::ASF::ambiguities($asf);

Returns a reference to an array of ambiguity reports in the ASF. The first and only argument must be an ASF object. The array returned will be be zero length if the parse was not ambiguous. Ambiguity reports are as described below.

While the "ambiguities()" method can be called to determine whether or not ambiguities exist, it is the more expensive way to do it. The $slr->ambiguity_metric() method tests an already-existing boolean and is therefore extremely fast. If you are simply testing for ambiguity, or if you can save time when you know that a parse is unambiguous, you will usually want to test for ambiguity with the "ambiguity_metric()" method before calling the "ambiguities()" method.

ambiguities_show()

  $actual_result =
    Marpa::R2::Internal::ASF::ambiguities_show( $asf, \@ambiguities );

Returns a string which contains a description of the ambiguities in its arguments. Takes two arguments, both required. The first is an ASF, and the second is a reference to an array of ambiguities, in the format returned by the ambiguities() method.

Major applications will often have their own customized ambiguity formatting routine, one which can formulate error messages based, not just on the names of the rules and symbols, but on knowledge of the role that the rules and symbols play in the application. This method is intended for applications which do not have their own customized ambiguity handling. For those which do, it can be used as a fallback for handling those reports that the customized method does not recognize or that do not need special handling. The format of the returned string is subject to change.

Symch reports

A symch report is issued whenever, in a top-down traversal of the ASF, an non-trivial symch is encountered. A symch report takes the form

   [ 'symch', $glade ]

where $glade is the ID of the glade with the symch ambiguity. With this and the accessor methods in this document, an application can report full details of the symch ambiguity.

Typically, when there is more than one kind of ambiguity in an input span, only one is of real interest. Symch ambiguities are usually of more interest than factorings. And if one ambiguity is uphill from another, the downhill ambiguity is usually a side effect of the uphill one and of little interest.

Accordingly, if a glade has both a symch ambiguity and a factoring ambiguity, only the symch ambiguity is reported. And if two ambiguities in the ASF overlap, only the one closest to the peak is reported.

Factoring reports

A symch report is issued whenever, in a top-down traversal of the ASF, an sequence of symbols is found which has more than one factoring. Factoring reports are specific -- they identify not just rules, but the specific sequences within the RHS which are differently factored -- multifactored stretches. Sequence rules especially have long stretches where the symbols are in sync with each other, broken by other stretches where they are out of sync. Marpa reports each of the ambiguous stretches. (A detailed definition of multifactored stretches is below.)

A factoring report takes the form

    [ 'factoring', $glade, $symch_ix, $factor_ix1, $factoring_ix2, $factor_ix2 ];

where $glade is the ID of the glade with the factoring ambiguity, and $symch_ix is the index of the symch involved. The multifactored stretch is described by two "identifying factors". Both factors are at the beginning of the stretch, and therefore have the same input start location. They differ in length.

The first of the two identifying factors has factoring index of 0, and its factor index is $factor_ix1. The second identifying factor has a factoring index of $factoring_ix2, and its factor index is $factor_ix2.

The identifying factors will usually be enough for error reporting, which is the usual application of these reports. Full details of the stretch are not given because they can be extremely large; are usually not of interest; and can be determined by following up on the information in the factoring report using the accessor methods described in this document.

Ambiguities in rules and symbols downhill from an ambiguously factored stretch are not reported. If a glade has both a symch ambiguity and a factoring ambiguity, only the symch ambiguity is reported.

The asf_to_basic_tree() code

  sub asf_to_basic_tree {
      my ( $asf, $glade ) = @_;
      my $peak = $asf->peak();
      return glade_to_basic_tree( $asf, $peak, [] );
  } ## end sub asf_to_basic_tree
  sub glade_to_basic_tree {
      my ( $asf, $glade, $seen ) = @_;
      return bless ["Glade $glade revisited"], 'My_Revisit'
          if $seen->[$glade];
      $seen->[$glade] = 1;
      my $grammar     = $asf->grammar();
      my @symches     = ();
      my $symch_count = $asf->glade_symch_count($glade);
      SYMCH: for ( my $symch_ix = 0; $symch_ix < $symch_count; $symch_ix++ ) {
          my $rule_id = $asf->symch_rule_id( $glade, $symch_ix );
          if ( $rule_id < 0 ) {
              my $literal      = $asf->glade_literal($glade);
              my $symbol_id    = $asf->glade_symbol_id($glade);
              my $display_form = $grammar->symbol_display_form($symbol_id);
              push @symches,
                  bless [qq{Glade $glade, Symbol $display_form: "$literal"}],
                  'My_Token';
              next SYMCH;
          } ## end if ( $rule_id < 0 )
          # ignore any truncation of the factorings
          my $factoring_count =
              $asf->symch_factoring_count( $glade, $symch_ix );
          my @symch_description = ("Glade $glade");
          push @symch_description, "Symch $symch_ix" if $symch_count > 1;
          push @symch_description, $grammar->rule_show($rule_id);
          my $symch_description = join q{, }, @symch_description;
          my @factorings = ($symch_description);
          for (
              my $factoring_ix = 0;
              $factoring_ix < $factoring_count;
              $factoring_ix++
              )
          {
              my $downglades =
                  $asf->factoring_downglades( $glade, $symch_ix,
                  $factoring_ix );
              push @factorings,
                  bless [ map { glade_to_basic_tree( $asf, $_, $seen ) }
                      @{$downglades} ], 'My_Rule';
          } ## end for ( my $factoring_ix = 0; $factoring_ix < $factoring_count...)
          if ( $factoring_count > 1 ) {
              push @symches,
                  bless [
                  "Glade $glade, symch $symch_ix has $factoring_count factorings",
                  @factorings
                  ],
                  'My_Factorings';
              next SYMCH;
          } ## end if ( $factoring_count > 1 )
          push @symches, bless [ @factorings[ 0, 1 ] ], 'My_Factorings';
      } ## end SYMCH: for ( my $symch_ix = 0; $symch_ix < $symch_count; ...)
      return bless [ "Glade $glade has $symch_count symches", @symches ],
          'My_Symches'
          if $symch_count > 1;
      return $symches[0];
  } ## end sub glade_to_basic_tree

The array_display() code

Because of the blessings in this example, a standard dump of the output array is too cluttered for comfortable reading. The following code displays the output from "asf_to_basic_tree()" in a more compact form. Note that this code makes no use of Marpa, and works for all Perl arrays. It is included for completeness, and as a simple example of array traversal.

    sub array_display {
        my ($array) = @_;
        my ( undef, @lines ) = @{ array_lines_display($array) };
        my $text = q{};
        for my $line (@lines) {
            my ( $indent, $body ) = @{$line};
            $indent -= 6;
            $text .= ( q{ } x $indent ) . $body . "\n";
        }
        return $text;
    } ## end sub array_display
    sub array_lines_display {
        my ($array) = @_;
        my $reftype = Scalar::Util::reftype($array) // '!undef!';
        return [ [ 0, $array ] ] if $reftype ne 'ARRAY';
        my @lines = ();
        ELEMENT: for my $element ( @{$array} ) {
            for my $line ( @{ array_lines_display($element) } ) {
                my ( $indent, $body ) = @{$line};
                push @lines, [ $indent + 2, $body ];
            }
        } ## end ELEMENT: for my $element ( @{$array} )
        return \@lines;
    } ## end sub array_lines_display

An alternative way of defining glade terminology

Here's a way of defining some of the above terms which is less intuitive, but more precise. First, define the glade length from glades A to glade B in an ASF as the number of glades on the shortest path from A to B, not including glade A. (Recall that paths are directional.) If there is no path between glades A and B, the glade length is undefined. Glade B is a downglade of glade A, and glade A is an upglade of glade B, if and only if the glade length from A to B is 1.

A glade A is uphill with respect to glade B, and a glade B is downhill with respect to glade A, if and only if the glade length from A to B is defined.

A peak of an ASF is a node without upglades. By construction of the ASF, there is only one peak. A glade with a token symch is trivial if it has no rule symches. A glade without a token symch is trivial if it has exactly one downglade.

The distance-to-peak of a glade "A" is the glade length from the peak to glade "A". Glade "A" is said to have a higher altitude than glade "B" if the distance-to-peak of glade "A" is less than that of glade "B". Glade "A" has a lower altitude than glade "B" if the distance-to-peak of glade "A" is greater than that of glade "B". Glade "A" has the same altitude as glade "B" if the distance-to-peak of glade "A" is equal to that of glade "B".

Cycles

In the current SLIF implementation, a forest is a directed acyclic graph (DAG). (In the mathematical literature a DAG is also called a "tree", but that use is confusing in the present context.) The underlying Marpa algorithm allows parse trees with cycles, and someday the SLIF probably will as well. When that happens, ASF's will no longer be "acyclic" and therefore will no longer be DAG's. This document talks about ASF's as if that day had already come -- it assumes that the ASF's might contain cycles.

In an ASF that contains one or more cycles, the concepts of uphill and downhill become much less useful for describing the relative positions of glades. For example, if glade A cycles back to itself through glade B, then

• Glade A will be uphill from glade B, and
• Glade B will be uphill from glade A; so that
• Glade B will be downhill from glade A, and
• Glade A will be downhill from glade B; and
• Glade A will be both downhill and uphill from itself; and
• Glade B will be both downhill and uphill from itself.

ASF's will always be constructed so that the peak has no upglades. Because of this, the peak can never be part of a cycle. This means that altitude will always be well defined in the sense that, for any two glades "A" and "B", one and only one of the following statements will be true:

• Glade "A" is lower in altitude than glade "B".
• Glade "A" is higher in altitude than glade "B".
• Glade "A" is equal in altitude to glade "b".

Token symches

In the current SLIF implementation, a symbol is always either a token or the LHS of a rule. This means that any glade that contains a token symch cannot contain any rule symches. It also means that any glade that contains a rule symch will not contain a token symch.

However, the underlying Marpa algorithm allows LHS terminals, and someday the SLIF probably will as well. This document is written as if that day has already come, and describes glades as if they could contain both rule symches and a token symch.

Maximum symches per glade

Above, the point is made that the number of symches in a glade, even in the worst case, is a very manageable number. For a particular case, it is not hard to work out the exact maximum. Here are the details.

There can be at most one token symch. There can be only rule symch for every rule. In addition, all rules in a glade must have the glade symbol as their LHS. Let the number of rules with the glade symbol on their LHS be "r". The maximum number of symches in a glade is "r+1".

Multifactored stretches

Marpa locates factoring ambiguities, not just by rule, but by RHS symbol. It finds multifactored stretches, input spans where a sequence of symbols within the RHS of a rule have multiple factorings. A multifactored stretch will sometimes encompass the entire RHS of a rule. In other cases, the RHS of a single rule might contain many multifactored stretches. This is often the case with sequence rules. Sequence rules can have a very long RHS, and in those situations narrowing down factoring ambiguities to specific input spans is necessary for precise error reporting.

The main body of this document worked with an intuitive "know one when I see one" idea of multifactored stretches. The exact definition follows. First we will need a series of preliminary definitions.

Consider the case of a arbitrary rule symch. Intuitively, a factoring position is a location within the factors of one of the factorings of that symch. It can be seen as a duple "<factoring_ix, factor_ix>" where "<factoring_ix>" is the index of a factoring within the symch, and "<factor_ix>" is the index of one of the factors of the factoring.

Let "SP" be a function that maps the symch's set of factoring indexes to the non-negative integers, such that for a factoring index "i" and factor index "j", "SP(i)=j", "j" is a valid factor index within the factoring "i". The function "SP" can be called a symch position.

Every symch position is equivalent to a set of factoring positions. The initial symch position is the symch position all of whose factoring positions have a factor index of 0. Equivalently, it is the constant function "ISP", where "ISP(i)=0" for all factoring indexes "i".

The factor with index "factor_ix" in the factoring with index "factoring_ix" is said to be the factor at factoring position "<factoring_ix, factor_ix>". A factor is one of the factors of a symch position if and only if it is a factor at one of its factoring positions.

An aligned symch position is a factoring position all of whose factors have the same start location. The location of an aligned symch position is that start location. The initial symch position is always an aligned factoring position. A synced symch position is an aligned symch position all of whose factors have the same length and symbol ID. A unsynced symch position is an aligned symch position that is not a synced symch position.

We are now in a position to define a multifactored stretch. Intuitively, a multifactored stretch is a longest possible input span that contains at least one unsynced symch position, but no synced symch positions. More formally, a multifactored stretch of a symch is a span of start locations within that symch, such that:

• Its first location is the location of unsynced symch position.
• Its first location is the initial symch position, or the first symch positiion after a synched symch position.
• Its end location is the end location of the symch, or a synced symch position, whichever occurs first.

Note that multifactored stretch are aligned in terms of input locations, but they do not have to be aligned in terms of factor indexes. The factoring positions of a multifactored stretch can have many different factor indexes. This is true of all rules, but it is particularly likely for a sequence rule, where the RHS consists of repetitions of a single symbol.