Editing Propositional formula (section)

== Propositional formula with "feedback" ==

The notion of a propositional formula appearing as one of its own variables requires a formation rule that allows the assignment of the formula to a variable. In general there is no stipulation (either axiomatic or truth-table systems of objects and relations) that forbids this from happening.<ref>McCluskey comments that "it could be argued that the analysis is still incomplete because the word statement "The outputs are equal to the previous values of the inputs" has not been obtained"; he goes on to dismiss such worries because "English is not a formal language in a mathematical sense, [and] it is not really possible to have a ''formal'' procedure for obtaining word statements" (p. 185).</ref>

The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p &or; s) = q, then let p = q. Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus impredicative.
Either of two conditions can result:<ref>More precisely, given enough "loop gain", either oscillation or memory will occur (cf McCluskey p. 191-2). In abstract (idealized) mathematical systems adequate loop gain is not a problem.</ref> oscillation or memory.

It helps to think of the formula as a [[black box]]. Without knowledge of what is going on "inside" the formula-"box" from the outside it would appear that the output is no longer a [[Function (mathematics)|function]] of the inputs alone. That is, sometimes one looks at q and sees 0 and other times 1. To avoid this problem one has to know the state (condition) of the "hidden" variable p inside the box (i.e. the value of q fed back and assigned to p). When this is known the apparent inconsistency goes away.

To understand [predict] the behavior of formulas with feedback requires the more sophisticated analysis of [[sequential circuit]]s. Propositional formulas with feedback lead, in their simplest form, to state machines; they also lead to memories in the form of Turing tapes and counter-machine counters. From combinations of these elements one can build any sort of bounded computational model (e.g. [[Turing machine]]s, [[counter machine]]s, [[register machine]]s, [[Macintosh computer]]s, etc.).

=== Oscillation ===

In the abstract (ideal) case the simplest oscillating formula is a NOT fed back to itself: ~(~(p=q)) = q. Analysis of an abstract (ideal) propositional formula in a truth-table reveals an inconsistency for both p=1 and p=0 cases: When p=1, q=0, this cannot be because p=q; ditto for when p=0 and q=1.

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[[File:Propositional formula oscillator 1.png|400px|thumb|right]]

'''Oscillation with delay''': If a delay<ref>The notion of delay and the principle of local causation as caused ultimately by the speed of light appears in Robin Gandy (1980), "Church's thesis and Principles for Mechanisms", in J. Barwise, H. J. Keisler and K. Kunen, eds., ''The Kleene Symposium'', North-Holland Publishing Company (1980) 123-148. Gandy considered this to be the most important of his principles: "Contemporary physics rejects the possibility of instantaneous action at a distance" (p. 135). Gandy was [[Alan Turing]]'s student and close friend.</ref> (ideal or non-ideal) is inserted in the abstract formula between p and q then p will oscillate between 1 and 0: 101010...101... ''ad infinitum''. If either of the delay and NOT are not abstract (i.e. not ideal), the type of analysis to be used will be dependent upon the exact nature of the objects that make up the oscillator; such things fall outside mathematics and into engineering.

Analysis requires a delay to be inserted and then the loop cut between the delay and the input "p". The delay must be viewed as a kind of proposition that has "qd" (q-delayed) as output for "q" as input. This new proposition adds another column to the truth table. The inconsistency is now between "qd" and "p" as shown in red; two stable states resulting:

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! style="font-weight:bold" | (
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=== Memory ===
[[File:Propositional formula flip flops 1.png|400px|thumb|right| About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output "q" feeding back into "p". The next simplest is the "flip-flop" shown below the once-flip. Analysis of these sorts of formulas can be done by either cutting the feedback path(s) or inserting (ideal) delay in the path. A cut path and an assumption that no delay occurs anywhere in the "circuit" results in inconsistencies for some of the '''total states''' (combination of inputs and outputs, e.g. (p=0, s=1, r=1) results in an inconsistency). When delay is present these inconsistencies are merely '''transient''' and expire when the delay(s) expire. The drawings on the right are called [[state diagram]]s.]]

[[File:Propositional formula 3.png|400px|thumb|right| A "clocked flip-flop" memory ("c" is the "clock" and "d" is the "data"). The data can change at any time when clock c=0; when clock c=1 the output q "tracks" the value of data d. When c goes from 1 to 0 it "traps" d = q's value and this continues to appear at q no matter what d does (as long as c remains 0).]]

Without delay, inconsistencies must be eliminated from a truth table analysis. With the notion of "delay", this condition presents itself as a momentary inconsistency between the fed-back output variable q and p = q<sub>delayed</sub>.

A truth table reveals the rows where inconsistencies occur between p = q<sub>delayed</sub> at the input and q at the output. After "breaking" the feed-back,<ref>McKlusky p. 194-5 discusses "breaking the loop" and inserts "amplifiers" to do this; Wickes (p. 118-121) discuss inserting delays. McCluskey p. 195ff discusses the problem of "races" caused by delays.</ref> the truth table construction proceeds in the conventional manner. But afterwards, in every row the output q is compared to the now-independent input p and any inconsistencies between p and q are noted (i.e. p=0 together with q=1, or p=1 and q=0); when the "line" is "remade" both are rendered impossible by the Law of contradiction ~(p & ~p)). Rows revealing inconsistencies are either considered [[transient state]]s or just eliminated as inconsistent and hence "impossible".

==== Once-flip memory ====
About the simplest memory results when the output of an OR feeds back to one of its inputs, in this case output "q" feeds back into "p". Given that the formula is first evaluated (initialized) with p=0 & q=0, it will "flip" once when "set" by s=1. Thereafter, output "q" will sustain "q" in the "flipped" condition (state q=1). This behavior, now time-dependent, is shown by the [[state diagram]] to the right of the once-flip.

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|}

==== Flip-flop memory ====
The next simplest case is the "set-reset" [[Flip-flop (electronics)|flip-flop]] shown below the once-flip. Given that r=0 & s=0 and q=0 at the outset, it is "set" (s=1) in a manner similar to the once-flip. It however has a provision to "reset" q=0 when "r"=1. And additional complication occurs if both set=1 and reset=1. In this formula, the set=1 ''forces'' the output q=1 so when and if (s=0 & r=1) the flip-flop will be reset. Or, if (s=1 & r=0) the flip-flop will be set. In the abstract (ideal) instance in which s=1 ⇒ s=0 & r=1 ⇒ r=0 simultaneously, the formula q will be indeterminate (undecidable). Due to delays in "real" OR, AND and NOT the result will be unknown at the outset but thereafter predicable.

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|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 0
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|style="background-color:#F2F2F2" | 
|style="background-color:#FDE9D9" | 1
|style="background-color:#F2F2F2" | state 1 with s & r simultaneously 1
|}

==== Clocked flip-flop memory ====
The formula known as "clocked flip-flop" memory ("c" is the "clock" and "d" is the "data") is given below. It works as follows: When c = 0 the data d (either 0 or 1) cannot "get through" to affect output q. When c = 1 the data d "gets through" and output q "follows" d's value. When c goes from 1 to 0 the last value of the data remains "trapped" at output "q". As long as c=0, d can change value without causing q to change.

* Examples
*# ( ( c & d ) &or; ( '''p''' & ( ~( c & ~( d ) ) ) ) = '''q''', but now let p = q:
*# ( ( c & d ) &or; ( '''q''' & ( ~( c & ~( d ) ) ) ) = '''q'''

The state diagram is similar in shape to the flip-flop's state diagram, but with different labelling on the transitions.

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! style="font-weight:bold" | Description
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| 0
| 1
| 0
| 
| 
| 0
|style="background-color:#DBE5F1" | 0
| 1
| 
|style="background-color:#FDE9D9" | 0
| 
|style="background-color:#FDE9D9" | 0
|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 1
| 
| 
| 0
|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 0
| 
| 1
| 
| 
| 
| 
| 
|style="background-color:#FDE9D9" | 0
| state 0 with ( d=1 & r=0 ), 0 is trapped
|- style="font-size:9pt" align="center"
|style="background-color:#BFBFBF;font-weight:bold" Height="12" | 3
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:red" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:red" | 0
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#D8D8D8" | q & p inconsistent
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2;font-weight:bold" Height="12" | 4
| 1
| 0
| 0
| 
| 
| 0
|style="background-color:#DBE5F1" | 0
| 0
| 
|style="background-color:#FDE9D9" | 1
| 
|style="background-color:#FDE9D9" | 1
|style="background-color:#DBE5F1" | 1
|style="background-color:#EAF1DD" | 1
| 
| 
| 0
|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 1
| 
| 0
| 
| 
| 
| 
| 
|style="background-color:#FDE9D9" | 1
| state 1 with (d =0 & c=0 ),  1 is trapped
|- style="font-size:9pt" align="center"
|style="background-color:#BFBFBF;font-weight:bold" Height="12" | 5
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 
|style="background-color:red" | 0
|style="background-color:#BFBFBF" | 
|style="background-color:red" | 1
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 1
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 0
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#BFBFBF" | 
|style="background-color:#D8D8D8" | q & p inconsistent
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2;font-weight:bold" Height="12" | 6
| 1
| 1
| 0
| 
| 
| 0
|style="background-color:#DBE5F1" | 0
| 1
| 
|style="background-color:#FDE9D9" | 1
| 
|style="background-color:#FDE9D9" | 1
|style="background-color:#DBE5F1" | 1
|style="background-color:#EAF1DD" | 1
|style="background-color:#EAF1DD" | 
| 
| 0
|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 0
| 
| 1
| 
| 
| 
| 
| 
|style="background-color:#FDE9D9" | 1
| state 1 with (d =1 & c=0 ), 1 is trapped
|- style="font-size:9pt" align="center"
|style="background-color:#F2F2F2;font-weight:bold" Height="12" | 7
| 1
| 1
| 1
| 
| 
|style="background-color:#FCFF7F" | 1
|style="background-color:#DBE5F1" | 1
|style="background-color:#FCFF7F" | 1
| 
|style="background-color:#FDE9D9" | 1
| 
|style="background-color:#FDE9D9" | 1
|style="background-color:#DBE5F1" | 1
|style="background-color:#EAF1DD" | 1
| 
| 
|style="background-color:#FCFF7F" | 1
|style="background-color:#DBE5F1" | 0
|style="background-color:#EAF1DD" | 0
| 
|style="background-color:#FCFF7F" | 1
| 
| 
| 
| 
| 
|style="background-color:#FDE9D9" | 1
| state 1 with ( d=1 & c=1 ):<br/>q=1 is following d=1
|}