Dotted rules¶

The idea behind Earley's algorithm is that you can parse by building a table of rules and where you are in those rules. "Where" means two things: location in the rule relative to the rule's symbols, and location relative to the parse's input stream.

Let's look at an example of a rule in a context-free grammar. Here's the rule for assignment from the Perl distribution's "perly.y"

"    termbinop -> term ASSIGNOP term"

"ASSIGNOP" is "perly.y"'s internal name for the assignment operator. In plain Perl terms, this is the ""="" character.

In parsing this rule, we can be at any of four possible locations. One location is at the beginning, before all of the symbols. The other three locations are immediately after each of the rule's three symbols.

Within a rule, position relative to the symbols of the rule is traditionally indicated with a dot. In fact, the symbol-relative rule position is very often called the dot location. Taken as a pair, a rule and a dot location are called a dotted rule.

Here's our rule with a dot location indicated:

"    termbinop -> X term ASSIGNOP term"

The dot location in this dotted rule is at the beginning. A dot location at the beginning of a dotted rule means that we have not recognized any symbols in the rule yet. All we are doing is predicting that the rule will occur. A dotted rule with the dot before all of its symbols is called a prediction or a predicted rule.

Here's another dotted rule:

"    termbinop -> term X ASSIGNOP term"

In this dotted rule, we are saying we have seen a "term", but have not yet recognized an "ASSIGNOP".

There's another special kind of dotted rule, a completion. A completion (also called a completed rule) is a dotted rule with the dot after all of the symbols. Here is the completion for the rule that we have been using as an example:

"    termbinop -> term ASSIGNOP term X"

A completion indicates that a rule has been fully recognized.

Earley items¶

The dotted rules contain all but one piece of the information that Earley's algorithm needs to track. The missing piece is the second of the two "wheres": where in the input stream. To associate input stream location and dotted rules, Earley's algorithm uses what are now called Earley items.

A convenient way to think of an Earley item is as a triple, or 3-tuple, consisting of dotted rule, origin and current location. The origin is the location in the input stream where the dotted rule starts. The current location (also called the dot location) is the location in the input stream which corresponds to the dot position.

Two noteworthy consequences follow from the way in which origin and current location are defined. First, if a dotted rule is a prediction, then origin and current location will always be the same. Second, the input stream location where a rule ends is not tracked unless the dotted rule is a completion. In other cases, an Earley item does not tell us if a rule will ever be completed, much less at which location.

Progress report lines¶

The last field of each Progress Report line shows, in fully expanded form, the dotted rule we were working on. Since that is the most important information, it may be tempting to skip the rest of this section, and move directly forward with the debugging.

In fact, you might want to do exactly that -- skip to the beginning of the next section. What follows talks about the details of the format of the first few fields in each progress report line. These first few fields, while helpful, are also usually one or more of obvious in their meaning, not relevant to our example, and repetitive of information which can be deduced from other fields.

      F0 @0-3 Expression -> Factor .

Prefixed to the dotted rule are two fields: ""F0 @0-3"". The ""F0"" says that this is a completed or final rule, and that it is rule number 0. The rule number is used in other tracing and debugging output, when displaying the whole rule would take too much space. In what follows we won't need the rule number.

The ""@0-3"" describes the location of the dotted rule in the parse. In its simplest form, the location field is two location numbers, separated by a hyphen. The first location number is the origin, the place where Marpa first started recognizing the rule. The last location number is the dot location, the location location of the dot in a dotted rule. ""@0-3"" say that this rule began at location 0, and that the dot is at location 3.

The current location is location 3, and this is no coincidence. Whenever we are displaying the progress report for an location, all the progress report lines will have their dot location at that location.

As an aside, notice that the left hand side symbol is "Expression". That is the start symbol. The presence of a completed start rule in our progress report indicates that if our input ended at location 3, it would be a valid sentence in the language of our grammar.

Let's look at another progress report line:

      R4:1 x2 @0,2-3 Factor -> Factor . Multiply Factor

Here the ""R4:1"" indicates that this is rule number 4 (the ""R"" stands for rule number) and that its dot position is after the first symbol on the right hand side. Symbol positions are numbered using the ordinal of the symbol just before the position. Symbols are numbered starting with 1, and symbol position 1 is the position immediately after symbol 1.

The next field (""x2"") is new. It is a count. A progress report can contain multiple instances of the same dotted rule, and when there is more than one, a count field is included in the progress report line. Here the ""x2"" indicates that there are two instances of "Factor -> Factor . Multiply Factor" at this location.

Multiple instances of a dotted rule will differ in their origin, and where they do, this is shown in the location field of the progress report line. Here the location field is ""@0,2-3"", which indicates that one instance of this dotted rule has its origin at location 0, and the other has its origin at location 2. All instances reported on a single progress report line will always have the same dot location, and in this case it is location 3.

Predicted rules also appear in progress reports:

    P2 @2-2 Factor -> . Number

Here the ""P"" in the summary field means "predicted". As with much of the information in the summary field, this only repeats what is obvious from the full expansion of the dotted rule later in the line. But final (or completed) and predicted rules can be important and the initial "F" and "P" make these lines easy to spot.

Notice that in the predicted rule, the origin is the same as the dot location. This will always be the case with predicted rules.

For any given location, no predicted rule has more than one instance. For other dotted rules, there may be many instances of the dotted rule at a single location. In grammars with right recursion, the number of instances is limited only by the length of the recursion. The length of a recursion is limited primarily by the available memory.

When there are many instances of a dotted rule at a single location, it is inconvenient to show all the origins in a comma-separated list. In that case the origins in the location field are shown as a range, with the earliest separated from the most recent by a ""..."". The example in this document contains no lines with a large number of instances, but here is an example from another grammar. This is the progress report line for the completed rule in a right recursion of length 20.

    F1 x20 @0...19-20 Top_sequence -> Top Top_sequence .

OK! Now to find the bug¶

Here again are progress reports at the location where things went wrong:

      F0 @0-3 Expression -> Factor .
      F2 @2-3 Factor -> Number .
      R4:1 x2 @0,2-3 Factor -> Factor . Multiply Factor
      F4 @0-3 Factor -> Factor Multiply Factor .

We see that we have completed rules for "Expression", and "Factor", as expected. We also see two Earley items that show that we are in the process of building another "Factor", and that it is expecting a "Multiply" symbol. This is not the rule we want, but it explains why the "trace_terminals" output showed that the recognizer was expecting a "Multiply" symbol.

What we want to know is, why is the recognizer not expecting an "Add" symbol? Looking back at the grammar, we see that only one rule uses the "Add" symbol: the rule ""Term -> Term Add Term"". The next step is to look at the Earley items for this rule. But there is a problem. We don't find any.

Next, we ask ourselves, what is the earliest place the ""Term -> Term Add Term"" rule should be appearing? The answer is that there should be a prediction of ""Term -> Term Add Term"" at location 0. So we look at the predictions at location 0.

      P0 @0-0 Expression -> . Factor
      P2 @0-0 Factor -> . Number
      P4 @0-0 Factor -> . Factor Multiply Factor

No ""Term -> Term Add Term"" rule. We are never even predicting a ""Term -> Term Add Term"" rule. We look back at the grammar, and start from the beginning.

    { lhs     => 'Expression', rhs => [qw/Factor/] },
    { lhs => 'Term',       rhs => [qw/Factor/] },
    { lhs => 'Factor',     rhs => [qw/Number/] },
    {   lhs    => 'Term',
        rhs    => [qw/Term Add Term/],
        action => 'do_add'
    },
    {   lhs    => 'Factor',
        rhs    => [qw/Factor Multiply Factor/],
        action => 'do_multiply'
    },

The start symbol is "Expression" and we do see a rule with "Expression" on the left hand side. "Expression" in turn produces a "Factor" symbol, and there are two rules with "Factor" on the left hand side.

But none of these rules ever produce a "Term". In fact, however far we follow the productions, no rule ever produces a "Term". At this point we see the problem: If we start at the start symbol, and follow the rules of our grammar, we will never get to a "Term" symbol. Which is exactly what that first warning message was saying.

Now that we know what is wrong, we can reread our grammar, and see that our "Expression -> Factor" rule is wrong. It should be "Expression -> Term". Change that and the problem is fixed.

progress()¶

    my $report0 = $recce->progress(0);

    my $latest_report = $recce->progress();

Given the parse location (Earley set ID) as its argument, the "progress()" recognizer method returns a reference to an array of "report items". The parse location may be negative. An argument of -X will be interpreted as location N+X+1, where N is the latest Earley set. This means that an argument of -1 indicates the latest Earley set, an argument of -2 indicates the Earley set just before the latest one, etc.

Each report item is a triple: an array of three elements. The three elements are, in order, rule ID, dot position, and origin. The data returned by the two displays above, as well as the data for the other parse locations in our example, are shown below.

The rule ID is the same number that Marpa uses to identify rules in tracing and debugging output. Given a rule ID, an application can find its LHS and RHS symbols using the grammar's "rule()" method.

Dot position is -1 for completions, and 0 for predictions. Where the report item is not for a completion or a prediction, dot position is N, where N is the number of RHS symbols successfully recognized at the parse location of the progress report.

Origin is parse location (Earley set ID) at which the rule application reported by the report item began. For a prediction, origin will always be the same as the location of the parse report.

Progress reports and efficiency¶

When progress reports are used for production parsing, instead of just for debugging and tracing, efficiency considerations become significant. We will start with the good news. "progress()" is implemented in C, so that the application can usually expect calls to "progress()" to be extremely fast.

Now, as to the potential bad news: Applications planning frequent use of calls to "progress()" need to be aware that, where there is a right recursion at a parse location, "progress()" needs to follow the entire chain of recursions to create the progress report. That this is happening will always be evident from the parse report itself, which will contain one report item for every completion in the right-recursive chain of completions. If an application tries to produce progress reports for a large number of parse locations, and these parse locations have long right recursive chains, it can prove computationally expensive.

The efficiency consideration just mentioned for following right recursions is not an issue for left recursions. Left recursions only produce at most two report items per parse location and are extremely fast to process. It is also not an issue for Marpa's sequence rules, because sequence rules are implemented internally as left recursions.

Code¶

    my $grammar = Marpa::R2::Grammar->new(
        {   start          => 'Expression',
            actions        => 'My_Actions',
            default_action => 'first_arg',
            rules          => [
                ## This is a deliberate error in the grammar
                ## The next line should be:
                ## { lhs => 'Expression', rhs => [qw/Term/] },
                ## I have changed the Term to 'Factor' which
                ## will cause problems.
                { lhs => 'Expression', rhs => [qw/Factor/] },
                { lhs => 'Term',       rhs => [qw/Factor/] },
                { lhs => 'Factor',     rhs => [qw/Number/] },
                {   lhs    => 'Term',
                    rhs    => [qw/Term Add Term/],
                    action => 'do_add'
                },
                {   lhs    => 'Factor',
                    rhs    => [qw/Factor Multiply Factor/],
                    action => 'do_multiply'
                },
            ],
        }
    );

    $grammar->precompute();
    my @tokens = (
        [ 'Number',   42 ],
        [ 'Multiply', q{*} ],
        [ 'Number',   1 ],
        [ 'Add',      q{+} ],
        [ 'Number',   7 ],
    );
    sub My_Actions::do_add {
        my ( undef, $t1, undef, $t2 ) = @_;
        return $t1 + $t2;
    }
    sub My_Actions::do_multiply {
        my ( undef, $t1, undef, $t2 ) = @_;
        return $t1 * $t2;
    }
    sub My_Actions::first_arg { shift; return shift; }
    my $recce = Marpa::R2::Recognizer->new(
        { grammar => $grammar, trace_terminals => 2 } );
    my $token_ix = 0;
    TOKEN: for my $token_and_value (@tokens) {
        last TOKEN if not defined $recce->read( @{$token_and_value} );
    }
    $progress_report = $recce->show_progress( 0, -1 );

Trace output¶

    Inaccessible symbol: Add
    Inaccessible symbol: Term
    Setting trace_terminals option
    Expecting "Number" at earleme 0
    Accepted "Number" at 0-1
    Expecting "Multiply" at 1
    Accepted "Multiply" at 1-2
    Expecting "Number" at 2
    Accepted "Number" at 2-3
    Expecting "Multiply" at 3
    Rejected "Add" at 3-4

Note the use of the term "earleme". If you are using the default input model, you can assume that earleme means "location": the earleme and the location will always be exactly the same. Advanced users, using alternative input models, may set it up so that earleme and location are two different things, and in that case the distinction will matter.

show_progress() output¶

    P0 @0-0 Expression -> . Factor
    P2 @0-0 Factor -> . Number
    P4 @0-0 Factor -> . Factor Multiply Factor
    F0 @0-1 Expression -> Factor .
    F2 @0-1 Factor -> Number .
    R4:1 @0-1 Factor -> Factor . Multiply Factor
    P2 @2-2 Factor -> . Number
    P4 @2-2 Factor -> . Factor Multiply Factor
    R4:2 @0-2 Factor -> Factor Multiply . Factor
    F0 @0-3 Expression -> Factor .
    F2 @2-3 Factor -> Number .
    R4:1 x2 @0,2-3 Factor -> Factor . Multiply Factor
    F4 @0-3 Factor -> Factor Multiply Factor .

progress() outputs¶

These section contains the output of the "progress()" method -- the progress reports in their "raw" format. The output is shown in Data::Dumper format, with "Data::Dumper::Indent" set to 0 and "Data::Dumper::Terse" set to 1.

The "Data::Dumper" output from "progress()" at location 0:

    [[0,0,0],[2,0,0],[4,0,0]]

The "Data::Dumper" output from "progress()" at location 1:

    [[0,-1,0],[2,-1,0],[4,1,0]]

The "Data::Dumper" output from "progress()" at location 2:

    [[2,0,2],[4,0,2],[4,2,0]]

The default "progress()" output is for the latest Earley set, which is location 3 in our example. Here is the "progress()" output for location 3.

    [[0,-1,0],[2,-1,2],[4,-1,0],[4,1,0],[4,1,2]]