How about using a bitmap state hash, deleting as you go to limit memory footprint. Single pass.....

my %states = (
    'begin'      => 1,
    'doing-work' => 2,
    'complete'   => 4,
);
my $re = join '|', keys %states;
$re = qr/^(\S+)\s*($re)/;

my %fail_decode = (
    6 => 'no begin',
    5 => 'no doing work',
    4 => 'no begin or doing work',
    3 => 'no complete',
    2 => 'no begin or complete',
    1 => 'no doing work or complete',
);

my %status;
while(<DATA>) {
    next unless /$re/;
    $status{$1} += $states{$2};
    delete $status{$1} if $status{$1} == 7;
}

printf "$_\t%s\n", $fail_decode{$status{$_}} for keys %status;

__DATA__
foo begin
bar begin
baz complete
foo doing-work
foo complete
bar doing-work
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