Bill,
There's a lot of ways to interpret your question... Yamaha and Technics use one simplified chord recognition scheme which doesn't require (or teach) musical logic, however it's limited to basic chord types. Korg, Roland, Gem, Ketron, etc. use a different scheme that recognizes partial chords and will supply the "missing" notes. However there are subtle variations among these brands. For example: Roland's "Chord Intelligence" will recognize Eb with C above as C-minor, but Korg reads this as Eb6 UNLESS you also press G. However, if on a Korg you play the Eb ABOVE C, just two notes will suffice. There are usually diagrams in the back of the user manuals illustrating which notes (at a minimum) must be pressed to signal each type of chord. In recent years, Korg and Roland have added a menu setting for the Yamaha style simplified chording scheme, but so far even Yamaha's advanced boards don't have a menu setting that emulates the Roland/Korg scheme.

Most boards also have a "professional" mode that requires 3 or 4 notes to be pressed before a chord is recognized. Even here, there are subtle but important differences which go beyond what's published in the manuals. For example, the more advanced Yamahas generally won't give you an Em7 unless you play it in root position. Other inversions get interpreted as G6 (with a usually-incorrect G bass.) To get around this, Casio's latest boards have a mode that essentially disables recognition of the 6th chord to give you the option of playing minor 7ths inverted without moving your hand. Roland and Korg also differ in their recognition of suspended chords in their various inversions. High-end Yamahas didn't used to recognize B,C,E as CM7. But beginning with the PSR-s900, now they do. Low-end Yamahas always permitted this "close" inversion! Etc., etc.

I don't want to reignite the earlier debate on here about "rootless" jazz voicings. Suffice to say Yamaha and Korg have provisions for recognizing these, although their repertoires differ, and to the chagrin of some, Yamaha revised their logic a few years after bringing this feature to market.

All of the truly pro boards have a provision for specifying the bass note to drive the arranger. The fingering varies by manufacturer and can get pretty tricky (sometimes spanning more than an octave!) Having a separate "Bass Inversion" button on the panel facilitates changing this setting on the fly, and effectively doubles the number of chord recognition modes. Yamaha and Korg now have hybrid modes where the lower of two keys pressed is recognized as the bass note, and three or more keys are interpreted as a "normal" chord. But both brands suffer from limitations and inconsistencies, IMO. If I had the patience to learn, I would imagine the cleanest approach to playing on-bass chords would be to use foot pedals, organ-style.

Some boards, such as Casio and Roland in "Standard" mode, acknowledge "free form" chords which will play a thin-sounding or even discordant accompaniment in response to open fourths or fifths. Others won't change the accompaniment at all unless at least three notes are played simultaneously. One handy feature that appeared on Roland's Prelude (and disappeared just as quickly on their current BK-5!) is a "chord latch," or momentary foot switch function that temporarily locks in the current chord, allowing you to play multiple notes in the left hand without fear of changing the chord. Ketron may have this feature also, I can't recall.

Roland, Korg, Ketron, etc. allow the user to create at least 3 different patterns within each variation generally corresponding to Major, Minor, and 7th chords. Reportedly some Yamaha factory styles exhibit this behavior, although a user can only create such styles for Yamaha by using 3rd-party software. Older Rolands had a feature called "Chord Family Assign" that allowed a user to specify which of the three patterns would be used for various "advanced" chords. Somehow the programming team overlooked this functionality when they developed the operating system for the G-70 and newer models.

The more advanced style formats have "transposition rules" that determine whether the recorded accompaniment pattern is transposed by scale degree ("parallel"), or whether an effort is made to sustain tones from the previous chord(s) ["Fixed" or "Nearest".] Also, whether additional notes ("tension") are sounded or ignored. Most brands have a setting called "retrigger" that determines whether the arranger will sound a note if a chord is changed between beats. Usually retrigger is a style creation parameter. However, Gem allows the user to specify it during playback, and Roland lacks a user-definable retrigger parameter. Experimentation suggests that Roland embeds it in the voice sample selected for each accompaniment part.

Another subtle point is how an arranger handles things when you don't get all of the keys down at the same time. To discern this behavior, you have to listen carefully or look at a MIDI recording of the accompaniment. Korg, especially the PA50/80 used a fairly large time window for chord input which makes glitches rare, although you have to lead the beat by a little more to compensate for the slight delay. Most recent Yamahas and the G70/E80 have a sort of "hair trigger" that recognizes a chord right away. Sometimes this can lead to glitching of the bass or the sounding of a false, premature chord. These kinds of glitches make music sound amateurish (arrangerish?) I don't sing, but I can just imagine what these false chords would do to the vocal harmony. I advocate that all arrangers should have a user-adjustable "window" from real time to about 60 milliseconds (.060 sec.), so the player can set the trade off between responsiveness and accuracy. If you're a little late with a key or two, Yamaha (and I think the recent Rolands) sneak the added notes in using pitch bends and portamento. These algorithms are tuned to sound good in local mode on the keyboard, but in certain cases I've noticed there is no MIDI 'note on' message, which limits interconnectivity with sound modules and other devices.

To sum up, what you're asking about gets into the very complex and defining heart of arranger operation. Perhaps also, into the differences between each manufacturer's proprietary technical approach. Hopefully others will chime in, and through discussion we can create a documented "knowledge base" of this behavior based on our collective experience. The MIDI standard doesn't attempt to address style data or arranger play. However, as each manufacturer has used proprietary technology to achieve a unique and realistic sound, consistency and inter-operability in this area have suffered. I totally agree that products from all manufacturers would benefit from more customizability, consistency, and inter-operability in this area. Good question Bill!!