yeah, how about not looping through the file more than once for starters.

#!perl -wl

use strict;

#bit fields
my %status = qw(
        begin           1
        doing-work      2
        complete        4
);
# figure out what a "perfect" score is
my $ok = (2 ** (scalar keys %status)) - 1;
# where we keep stuff
my (%entries);

#open F, "<logfile" or die $!;
while (<DATA>) {
        if (/^(\d{4}HA-\d{5}) (\S+)$/) {
                $entries{$1} |= $status{$2};
        }
}
#close F;

#filter out entries that passed
for (grep{ $entries{$_} == $ok  }keys %entries) {
        delete $entries{$_};
}

printf(
        "%d entries failed\n",
        scalar keys %entries
);
for (sort keys %entries) {
        print "$_: $entries{$_}";
}

__DATA__
1234HA-00001 begin
1234HA-00002 begin
1234HA-00003 begin
1234HA-00001 doing-work
1234HA-00002 doing-work
1234HA-00001 complete
[download]