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Complex State

model complex state as sets of facets, eg.:

  °door:          { open,      close, }
  °cooking:       { true,      false, }
  °mains:         { connected, disconnected, }
  °powerbutton:   { pressed,   released, }

from keys with various possible aspects follows a number of namespaced states:

  °powerbutton: { pressed, released, } =>
    { °powerbutton:pressed, °powerbutton:released, }

The powerset of all namespaced states lists all states that the model could in theory be in; of these, some, like

  °mains:disconnected and °cooking:true

will be physically impossible, so should be logically excluded from the states; others, such as

  °door:open and °cooking:true

are undesirable but, crucially, physically possible, and must likewise be logically excluded.

States are views as key/value pairs, where the key identifies a component of the physical machine, and the value encodes the position that component is in or the action that the component is performing.

Observe states may be (quasi-) continuous, such as °temparature:51C or °thermostatdial:60C; in such cases, a comparator °temparature < °thermostatdial or °temparature > °thermostatdial can be used to decided whether to heat, to switch off heating, to cool, or to switch off cooling, as the case may be.

Keys may also represent ongoing actions such as °cooking:{ongoing,stopped,interrupted}.

Conjunctions and Disjunctions

State vectors can be linked via boolean logic:

°door:closed    ∧   °door^open              =>  °door:open
°door:open      ∧   °door^open              =>  °door:open
°door:open      ∧   °plug^insert            =>  °door:open
°magnetron:on   ∧   °door^open              =>  °magnetron:off
°plug:loose     ∨   °powerbutton:released   =>  °powerlight:off
°door^open                                  =>  °magnetron:off

The disjunction ( or or operator or 'union') we can safely discard with as it is easily representable by inserting multiple transitions:

°plug:loose ∨ °powerbutton:released   => °powerlight:off
°plug:loose                           => °powerlight:off
°powerbutton:released                 => °powerlight:off

However, conjunctions ( or and operator or 'intersection') must still be explicitly expressed:

°plug:inserted ∧ °powerbutton:pressed  =>  °powerlight:on

There are two ways to capture this in (Postgre)SQL: either with arrays of values, or by grouping clauses by means of a term ID; this solution has the advantage that it leaves a natural opening for expressing assertion/negation, here called pred (for 'predicate'):

                  'condition'               'consequence'
                  'if'                      'then'
term        pred    source_item           =>  target_item
——————————— ——————— ————————————————————— ——— ———————————————————
term:20     T       °plug:inserted        =>  °powerlight:on
term:20     T       °powerbutton:pressed  =>  °powerlight:on
term:50     T       °plug:loose           =>  °powerlight:off
term:51     T       °powerbutton:released =>  °powerlight:off

It can be readily seen that in the above table

  • assuming there are only two states for the °plug (either :inserted or :loose), we can also write not °plug:inserted for °plug:loose and vice versa;
  • the term IDs for disjunct items must all be different (hence, 'dis'junction); and
  • the value for target_item must be the same for all rows referring to the same term.

This leads us to a generalization: what if we didn't use a target item but a target term? That would allow us to notate both consequences and conditions as vectors in a unified fashion. Moreover, let's introduce the concept of a 'phrase', which we define as the sequence of terms that lead from (conjunctions of) conditions (and optional intermediaries) to consequences. In the below table, we have added a nonsense term:99 to show that phrases may overlap in their consequences; this is the effect of disjunctions:

( a ) ∨ ( b ∧ c ) => d => ( e ∧ f )

holds when

( (   a   ) => d => ( e ∧ f ) )
( ( b ∧ c ) => d => ( e ∧ f ) )


We will also introduce two actions, °powerlight^on and °powerlight^off, to replace the states that we used in the earlier tables; this to express more clearly that, from the condition, a dynamic consequence followed, one that, despite appearances, has multiple consequences (namely, both turn on the power indicator and ring a bell).

                    'condition'               'consequence'
                    'if'                      'then'
term        pred    source_item           =>  target_item
——————————— ——————— ————————————————————— ——— ———————————————————
term:99     T       °foo^bar              =>  term:21
——————————— ——————— ————————————————————— ——— ———————————————————
term:20     T       °plug:inserted        =>  term:21
term:20     T       °powerbutton:pressed  =>  term:21
term:21     T       °powerlight^on        =>  term:22
term:22     T       °bell^chime               ∎
term:22     T       °powerlight:on            ∎
——————————— ——————— ————————————————————— ——— ———————————————————
term:50     T       °plug:loose           =>  term:52
term:51     T       °powerbutton:released =>  term:52
term:52     T       °powerlight^off       =>  term:53
term:53     T       °powerlight:off           ∎

One may note at this junction that the states of the °powerbutton component have not been modelled satisfactorily; after all, there should normally some kind of user interaction that is responsible for toggling its state. Regardless of whether we have a switch that requires flipping or pressing it, what the user does is reverse the state of switch by some kind of gesture; let's model this as a verb ^actuate. This, then, leads to the next refinement.

Observe that °powerbutton^actuate is like a public member of the microwave's 'API', as it were, whereas °powerlight^on and ^off are like private members in that they cannot be directly caused from outside of the automaton:

term        pred    source_item           =>  target_item
——————————— ——————— ————————————————————— ——— ———————————————————
term:10     T       °powerbutton^actuate  =>  term:11
term:10     T       °powerbutton:released =>  term:11
term:11     T       °powerbutton:pressed      ∎
term:12     T       °powerbutton^actuate  =>  term:14
term:12     T       °powerbutton:pressed  =>  term:14
term:14     T       °powerbutton:released     ∎
——————————— ——————— ————————————————————— ——— ———————————————————
term:20     T       °plug:inserted        =>  term:21
term:20     T       °powerbutton:pressed  =>  term:21
term:21     T       °powerlight^on        =>  term:22
term:22     T       °bell^chime               ∎
term:22     T       °powerlight:on            ∎

There's yet another problem apparent: in the chain °plug:inserted ∧ °powerbutton:pressed => °powerlight^on ∧ °bell^chime, no mention of time or rising vs. falling flanks is made; therefore, if we interpreted the phrase as being timeless, then the °bell should be ^chimeing all the time. This is probably not what the customer wants, an oven that rings all the time when being in use.

In order to have the bell just say 'pling', we insert a new internal verb, °powerbutton^press, that is then used to both toggle the switch and chime the bell; we note that this will only work if we can make it so that, on the one hand,

  • states (°x:y) are eternally valid but get overriden by later states with the same component,

and on the other,

  • events (°x^y) are exhausted as soon as all direct consequences have been retrieved.

Since we want the automaton to only process a single event in each cycle, that also implies further that

  • there are no truly simultaneous events: each event comes before or after any other, if any

meaning that in order to model conjunctions of events: °a^b ∧ °u^v => ..., we have to do so by having the events first cause a state change: °a^b => °c:d; °u^v => °w:x;, and only when those partial states do combine can a consequence happen: °c:d ∧ °w:x => .... So °switch^activate ∧ °plug^insert can never be fulfilled; this will, therefore, be ruled out by a higher-order regulation to ensure that

  • a phrase may only contain at most one event.

Instead, a more circumlocutionary suite like

°switch^activate                        =>  °switch:activated;
°plug^insert                            =>  °plug:inserted;
°switch:activated ∧ °plug:inserted      =>  °device^start`

must be used.

term        pred    source_item           =>  target_item
——————————— ——————— ————————————————————— ——— ———————————————————
term:10     T       °powerbutton^actuate  =>  term:11
term:10     T       °powerbutton:released =>  term:11
term:11     T       °powerbutton^press        ∎
term:11     T       °powerbutton:pressed      ∎
term:12     T       °powerbutton^actuate  =>  term:14
term:13     T       °powerbutton:pressed  =>  term:14
term:14     T       °powerbutton:released     ∎
term:14     T       °powerbutton^release      ∎
——————————— ——————— ————————————————————— ——— ———————————————————
term:20     T       °plug:inserted        =>  term:21
term:20     T       °powerbutton^press    =>  term:21
term:21     T       °powerlight^on        =>  term:22
term:22     T       °bell^chime               ∎
term:22     T       °powerlight:on            ∎

Continuous Values


  • °thermometer:temp < -18C
  • °bicycle:speed > 25kmh
  • °light:on = true

If / Then / Else Conditions



Turing Completeness


    °door         :closed
  + °plug         :inserted
  + °cooking      :stopped
  + °startbutton  :released
  + °startbutton  ^press
  = °button       :pressed
  + °cooking      :ongoing

  + °cooking      :ongoing
  + °heater       :on
  + °temparature  :toohigh
  = °heater       :off

  + °cooking      :ongoing
  + °heater       :off
  + °temparature  :toolow
  = °heater       :on
°door:closed °plug:inserted °cooking:ongoing °button^release  => °button:released °cooking:stopped


event:  °component^verb
state:  °component:aspect

The FlowMatic Finite Automaton

Symbolic and Built-In States

  • FIRST—the point of a RESET act; must be the 'point of origin' in the transition table.

  • LAST

  • *—a.k.a. 'star' or 'catchall tail'; a transition with a catchall tail will match any point, even a non-existing one when the journal is empty after after initialization and can thus be used to bootstrap the FA.

  • ...—a.k.a. 'ellipsis', 'anonymous', 'continuation' or 'continue'; may occur as point (where it signifies 'continue with next transition' in order of inseetion to transitions table) or as tail (where it means 'continued from previous transition').

Symbolic and Built-In Acts

  • RESET—to be called as the first act after setup; initializes journal (but not the board).

  • START—to be called as the first act of a new case.

  • STOP-to be called as the last act of a case.

  • ->—a.k.a. 'walkthrough', 'then' or 'auto-act'; indicates that the assciatiated command is to be executed without waiting for the next act. This allows to write sequences of commands.

Constraints on Transitions Table

  • All tuples ( tail, act ) must be unique; there can only be at most one transition to follow when the current state is paired with the upcoming act. The exception is the tuple ( '...', '->' ), which may occur any number of times.

  • A '*' (star) in the tail of a transition makes the associated act unique; IOW, a starred act can have only this single entry in the transitions table. Note It is possible that we re-define the star to mean 'default' transition for a given act in the future and lift this restriction.

  • A transition that ends in ... (continuation) must be followed by a transiton that starts with a ... (continuation); the inverse also holds. IOW, continuation points and tails must always occur in immediately adjacent lines of the transitions table.

  • For the time being, a transition with a continuation tail must have a walkthrough act. That is, a series of commands that are connected by ... (continuations) can not wait for a specific action anywhere; such series must always run to completion until a properly named point is reached.

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