Why not:

$pid = "/var/log/syslog.pid";
unless (stat($pid)) { print "$pid not running\n"; }
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Is there a reason why running stat on the pid file is a Bad Idea? I'm not suggesting it's the best way, mind. I'm just wondering if it's acceptable.

Updated: Thanks, bluto. I'd totally let that evacuate my brain. Cat the pid file to get the process id for which to grep in ps. Buh.

So, just to run mine into the ground:

$pid = "/var/run/syslogd.pid";
if (stat($pid)) {
        open(PID, "< $pid");
        my $id = <PID>;
        chomp $id;
        if($id) { print "$id running\n";
        } else { print "$pid not running\n"; }
        close PID;
} else {
        print "$pid not running\n";
}
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... but it's just for my own excercise at this point. The stat is pointless, since I could just as easily bail if it dies on opening the file in the file handle.