CO is used to connect "openers" to subjects of clauses:

             +----------CO------------+
             |                        |
1.        Apparently,                  they went to a movie
2.                 On Tuesday,                     they went to a movie
3.        Although they were tired,    they went to a movie
4.          Leaving the kids at home,  they went to a movie
5.        Abandoned by their parents,  they went to a movie
6.   Still upset about Joe,          they went to a movie

Various kinds of words have CO+ connectors: adverbs (ex.1),
prepositions (ex.2), conjunctions (ex.3), participles
(ex. 4-5), and adjectives (ex.6).

Openers may take commas; almost all words with CO+ therefore
have "({{Xd-} & Xc+} & CO+)". With participles and adjectives,
the comma is obligatory; "*Still upset about Joe they went to
a movie" seems wrong. The Xd- allows a comma before the phrase
as well as after. This frequently happens if the opener is not
at the beginning of the phrase: "They claimed that, on
Tuesday, they went to a movie." If the opener begins the
sentence, then a comma before the opener is of course
incorrect, but we allow it. See "X: Comma phrases".

Nouns have optional @CO- connectors, conjoined with their
C- and W- connectors, conjoined in turn with S+:

        dog: ...({({@CO-} & (W- or C-)) or R-} & S+) or O- or J-...)

(This expression is explained further below.)  Thus a CO link
always has the subject of a clause on the right.

Notice that the CO- on nouns carries "@": it is a
"multi-connector", allowing an indefinite number of
connections. In some cases, non-initial openers seems more
like interrupters than openers, but the parser usually treats
them as openers. In cases like this one, however, the second
opener may also be treated as a post-verbal modifier of the
first opener.

        On Tuesday, leaving their kids at home, apparently,
               they went to a movie

Openers on Different Kinds of Clauses
There are constraints on the way clauses can take
openers. Main clauses can always take openers (ex. 1 below),
as can subordinate and embedded clauses (ex. 2 and 3);
relative clauses can only do so if they involve a relative
pronoun (ex. 4 and 5); indirect questions cannot (ex. 6):

        1. Screaming furiously, Fred left the room
        2. I was about to leave when, screaming furiously,
                Fred attacked me
        3. I told her that, after the party, I would meet her
        4. This is the man who, in some ways, I would like to hire
        5. *This is a man, in some ways, I would like to hire
        6. *I wonder who, on Tuesday, Jane had lunch with

This is controlled by the fact that these clauses make
different kinds of connections. Main clauses make a W
connection to the left (usually to the wall); embedded and
subordinate clauses make a C connection (to a subordinating conjunction
or a verb or adjective). (In all these cases, the connection is made
through the subject of the clause.) Indirect-question subjects have no
left-branching connector at all. Relative clause subjects make
C connections when a relative pronoun is present; otherwise
they make R connections to the previous noun. Note that in the
above expression for "dog", C- and W- are conjoined with CO+;
R- is not.  Thus, when a noun subject is making a R-
connection backwards, or when it is making no connection at
all, it may not take an opener. Therefore ex. 5 and 6 are
rejected.

Questions and imperatives occasionally take openers, but this
is rare.  Openers on questions and imperatives are rejected by
our system; there is no way for the opener phrase to connect.

A further distinction is necessary here. Participle phrases
can modify main clauses as openers; they are very rarely
found, however, on dependent clauses (either embedded or
subordinate).

        Shouting loudly, Fred ran out of the room
        *Jane told us how shouting loudly, Fred ran out of the room
        *Jane left the party after shouting loudly, Fred ran
                out of the room

For this reason we give participle phrases COp+; we then create
two CO- connectors on nouns, one conjoined with W-, the other
with C-. The one with C- is subscripted COd-:

        {({@CO-} & W-) or ({@COd-} & C-) or R-} & S+

Participles as Openers
A further point is needed about participle openers. They
appear to take complements in the same manner as ordinary
participles. There is a problem here, however. While
participles normally make a connection to the left to an
auxiliary, they sometimes make a further-left connection to a
fronted object, for example:

              +-------B------+
              |    +---Pg----+
              |    |         |
        What book are you reading

For this reason, the Pg- or Pv- on participles must be to the left
of the complement expression on participles. However, openers may
not be used in this way. Moreover, on openers, the CO+
connector (directly disjoined with the Pg- or Pv-) must be
made further to the right than any complement connections:

           +----------CO---------+
           +---+                 |
           |   |                 |
        Saying he was innocent, John left the room
        Angered by what he saw, John left the room

           +------------------------+
           +--CO--+                 |
WRONG:     |      |                 |
       *Saying,  John left the room he was innocent
       *Angered, John left the room by what he saw

Here, then, the CO+ must be to the right of the complement
connectors on the expression. For this reason, we must fully
disjoin the CO+ and Pg+/Pv+ connectors on passive and progressive
participles.  This is necessary in any case for progressive
participles in the case of gerunds; for gerunds, also, the S+
must be to the right of the complement connectors on the
expression. (See "Ss#g".) We also directly disjoin COp+ here
with MVx- (used for comma participle phrases modifying verbs),
and MX*p- (used for comma participle phrases modifying nouns).

hitting: (Pg- & (O+...))   or    (O+...) & (Ss*g+ or COp+ or
                                    MVx- or MX*p-);

angered: (Pv- & {@MV+})    or    ({@MV+} & (COp+ or MVx- or MX*p-);

COq
COq is used for "paraphrasing" verbs, inserted as openers:
"After dinner, he said, he left." See "CP". It might be argued
that "he said" is really an "interrupter" here, not a opener:
it cannot be used unless there is another opener preceding
it. ("*He said, he left" seems wrong.) We treat it as an
opener; however, in this case we do require a comma both
before and after the paraphrasing expression, thus preventing
this false positive in another way.

COqi is used for paraphrasing verbs which require "filler-it"
as subject; see "SF: filler-it".