Time In Three Parts

A practical definition of ratio temporality

NOTE: This is an old version of this article. It has been updated. It now includes new content, such as information about how ratio-temporality is employed in biological neural networks. This inherently leads to an understanding of how ratio-temporality can be employed in artificial neural networks as well.

The newer version containing the new material about how ratio-temporality is employed in biological neural-networks, and in ANNs made with the Netlab development environment, is included in the Netlab book as Appendix C.

This online version will serve as a place to accumulate related resources and references.

Dominic John Repici 09-June-2009 (obsolete, newer version available) (C) Copyright 2004-2009 Dominic John Repici ALL RIGHTS RESERVED (v1.0m)


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This paper documents specific relationships between time, frequency, sequence, 
and temporality and uses that understanding of those relationships to propose 
an alternative narrow definition for the terms "temporal" and "temporality." 

Comments and feedback are welcome.

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What is "Temporal?"

Much like "consciousness," everybody seems to know what the word "temporal" means yet if you look around for a precise definition you may not be satisfied with what you find. Here's what the Oxford Dictionary says: "of or relating to time." The Merriam Webster online dictionary says: "a: of or relating to time as distinguished from space b: of or relating to the sequence of time or to a particular time" These are fine if all you need is "[it has something to do with] time." This paper represents my attempt to define a more precise sense of the word "temporal." My goal is to restrict the definition by explaining how temporal "relates to time" as well as its relationship to other components of a set loosely described by the phrase "relating to time." As defined here: Here are the more precise (restricted) definitions for the terms temporality and temporal proposed and supported in this paper. Temporality (noun): a: That component of time and/or space which is neither sequence or frequency. - That component of events in time and/or space that remains after sequence and frequency are removed. - The condition of being temporal. b: The relative distances or times between events or entities in time or space. Temporal (adj): a: Of or relating to that component of time and/or space which is neither sequence or frequency. b: Of or relating to the relative distances between events or entities in time or space.

Qualifier: Ratio-temporal

This paper does not propose replacing the existing definition of temporal. It only proposes a stricter sense of the term for when the specific sense described here is needed. For contexts where the word temporal is likely to be confused with its usual, more general definition, the sense described here may be qualified as "ratio-temporal." The reason for use of the word ratio as the qualifier for this stricter sense of temporal should become clear in this explanation.

Frequency Domain

Fourier series transforms and similar tools that follow, speak of a time domain and a frequency domain. Many mathematical tools now exist which can convert between these two domains. If you ask "How well does the tool perform the function for which it was designed?" the answer would usually be that the tool works exactly as it should. If you change the question to: "How well does the reversible function of the tool represent observations of real world phenomena?" you then find that information from the time domain is lost when you convert it to the frequency domain. Example 1: Events in time FIGURE 1: A chart showing pulses representing a set of events in time. There is a certain fractal nature to this (e.g., the 'up' and 'down' sides of each pulse are also events), but for now, we will just assume events at the pulse level. If you convert this from the time domain (shown) to the frequency domain, information is lost. That is, this exact waveform will not be returned when you convert it back to the time domain. Instead, the low- and high-frequency components will probably be imposed over the entire reconstituted waveform.

Relationships between domains

This is the most important point to understand from this demonstration. If you start with a representation in the frequency domain in the above example, you can convert it to the time domain and back as many times as you'd like without losing any of the original information. This reveals two things about the relationship between the time domain and the frequency domain. 1. The time domain "encompasses" or "includes" the entire frequency domain 2. The frequency domain does NOT include the entire time domain. In other words the relationship between the time and frequency domains is not an equal one. If represented diagrammatically the frequency domain is completely contained within the time domain (figure 3). /---------------\ | Time | | /---------\ | | |Frequency| | | \---------/ | \---------------/ -Figure 3- Expressed another way, we can say that frequency is a "component" of time. We also know that there are other components because at least one thing gets lost when we convert to frequency and back. In this case, we have seen that we lose SEQUENCE, and as we will see we also lose at least one other constituent as well (figure 4). /-------------------------------------------\ | Time | | /---------\ /-----------\ /---------\ | | |Frequency| |Sequence | |Others? | | | \---------/ \-----------/ \---------/ | \-------------------------------------------/ -Figure 4-

Representing Events In Time

As stated, this paper defines the temporal domain as a domain that exists within the context of events in time, that is outside of the frequency, or sequential components of the time domain. This relationship is shown diagrammatically in Figure 5. /-------------------------------------------\ | Time | | /---------\ /-----------\ /---------\ | | |Frequency| |Sequential | |Temporal | | | \---------/ \-----------/ \---------/ | \-------------------------------------------/ -Figure 5- As you will see, the relationship between the temporal domain and the time domain is similar to the relationship between the frequency domain and the time domain. This will be documented now in order to show that the relationships between time, frequency, sequential, and temporal domains diagrammed in figure 5 (above) are logically valid and usable for the purpose of supporting the proposed definition of temporality. This next illustration (Figure 6) will be used to relate the remainder of the concepts in this paper. It is a graphic representation of events along a time-line. event name Q G A X L time or space |---------------------------------------------> -Figure 6- This diagram depicts a series of five events occurring over a period of 41 equal intervals of time. It comprises frequency, sequence (order), and temporality. The amount of time each interval represents is not important; they could be milliseconds, years, or millennia. Likewise the events could be the amplitude levels of sine wave peaks or events in human history, the key is, it doesn't matter what the actual events or interval lengths are. The labels we've given are letters but should only be thought of as names for the events. They've even been placed out of letter order to further emphasize that no alphabetic order should be inferred. We could for example use pet names to label the entities. We could have also represented the low and high waveform amplitudes in example-1 as 'An' and 'Bn' respectively and spread them across the timeline accordingly. Again, the point is, context is unimportant. Only events over time are represented.

Sequential Domain

Sequence is that component of the time domain that embodies tense and causality, though temporality may share in contributing to the phenominon of causality. There are components of frequency, sequence, and temporality in the time line of Figure-6. As has been demonstrated above, converting to the frequency domain and back will shed sequence. Likewise, if you convert to the sequence domain and convert it back to the time domain you will be left with. QGAXL This shows that you lose frequency information (you will be left with a single frequency equal to the length of the sample interval). You also lose temporality as it is defined in this paper. Conversely, if we start from the sequence domain using the sequence QGAXL, we can convert it to the time-domain and back with no loss of information from the original (sequential) representation. This shows that just as in the case of the FREQUENCY domain, the SEQUENCE domain is a sub-set, fully encompassed by the time domain. This relationship is diagrammed in figures 4 and 5 (above).

Temporal Domain

Temporality is described here as the RATIOS of, or RELATIVE intervals between events. The temporal domain carries no information about frequency or sequence. Here's one way of representing the temporal domain from the time-line shown in figure 6: Q: G=2 A=3 X=7 L=8 G: Q=2 A=1 X=5 L=6 A: G=1 Q=3 X=4 L=5 X: Q=7 G=5 A=4 L=1 L: Q=8 G=6 A=5 X=1 -Figure 7- There are a variety of notational conventions we can use. To construct this notation I've counted the time intervals starting with the interval of the first event (Q) up to the interval just before the second event (G) which equaled 10 intervals. I continued on in similar fashion for each of the other events. For simplicity's sake, I've divided all numbers gleaned by five for the smallest whole number representation of the intervals. The resulting chart is represented in Figure 7. This chart (Figure 7) can be simplified quite a bit further but has been left completely filled in for clarity. But isn't this holding sequential information? No, the sequence is only revealed because of the way I've written it on the page. I don't have to write it in that order though. Here's the same chart of temporal information but presented without giving away any of the original sequence information. A: Q=3 L=5 X=4 G=1 G: L=6 Q=2 A=1 X=5 L: G=6 Q=8 X=1 A=5 X: Q=7 A=4 G=5 L=1 Q: A=3 G=2 X=7 L=8 -Figure 8- Ok, but frequency is preserved here isn't it? No. The only information carried in the temporal domain are the distances between events RELATIVE to the distances between other events (e.g. "There's twice as much time between A and X as there is between G and Q"). The actual intervals could be microseconds, years, or centuries among other things. The temporal representation retains no hint of this (it could even be miles or light-years). More to the point, the measured distance between any two events could be hours in one observation and microseconds or years in the next observation. As long as the RATIOS of measured distances between events remain the same, the temporal domain representation will remain the same. Figure 9A shows the temporal data converted back to the time domain. Note that it covers a shorter period. We haven't preserved the original sample period so the intervals themselves could represent a different amount of time as well. The arrow key example in figure 9C (below) may be a little confusing. When displayed on a terminal, put the cursor on the Q and use the arrow key to move to the G. It doesn't matter if you go left then down or down then left, it will be the same number of key-presses as you will use in the two diagrams above it. A: event name Q GA XL time or space |---------------------------------------------> B: It works backwards too: event name LX---AG-Q C: or this way: (arrow key -----Q presses ----G- on a ----A- keyboard) LX---- -Figure 9- This shows that, just as with the frequency and sequential domains, information from the other two component domains is lost when we convert from the time domain to the temporal domain and back. Also, if we start with only temporal information we can convert back and forth between it and the time domain without any loss of the original information. As we have already shown, the other two component domains share a similar relationship to the time domain. The relationships between the time domain and its three component domains can be represented as shown in figure 5 (above). Lastly, each of the frequency, sequential, and temporal domains do not seem to overlap with the information about the time domain contained within their two counterpart component domains.

Prediction Based On Temporal Ratios?

Is it possible to use temporal relationships to predict the timing of future events? For example, we experiment and observe three "related" events. That is, we make multiple observations, each with defferent absolute timings, but find that they always exhibit the same temporal ratio. X Y Z t|---------------------------------------------> Specifically, we note that they have this temporal component: (X=Y:4 Z:6 | Y=X:4 Z:2 | Z=X:6 Y:2). That is to say, period X<->Y is twice as long as period Y<->Z, and period X<->Z is 3 times as long as the interval Y<->Z, etc. Later we do more experiments and observe them again, perhaps with many different absolute time intervals, but always with the same temporal ratio: X Y Z t|---------------------------------------------> And again and again with the same temporal ratio: X YZ t|---------------------------------------------> X Y Z t|---------------------------------------------> The ratio relationships between the events should be observed to remain roughly the same over many observations, whether the absolute time-frame between X and Z spans a few seconds, a few hours, or a few decades. So, after many experiments we repeatedly see the same temporal ratio relationships when observing these three events. We establish a high probability that these are temporally related events.


In this case, having established a temporal relationship we can use observations of past event timing to predict when a future event will occur. NOW | <--Past | Future--> | X Y | Z t|---------------------------------------------> / Prediction (Y<->Z)=(X<->Y)/2 Again, whether the interval from the past (X<->Y) spans seconds, hours, or years, the interval spanning into the future can be predicted based on the previously observed ratio-temporal relationship between the intervals.

Prediction Based On Cause, Sans Sequence?

Is causality exclusively an attribute of sequence ? While causality is obviously embodied in sequence ("this follows that"), is it wrong to assume that sequence is the ONLY component supporting the notion of causality? Is there any possibility that cause-and-effect relationships exist between events in the ratio-temporal domain as defined here? That is, just as it can be assumed that one event can "cause" another event that follows it sequentially, are there periods between events that can be shown to "cause" time periods between other events? That was rambling at best in pros, perhaps I can draw a picture: To eliminate sequence from the prediction we made in the above section, we should look for similar ratio-temporal relationships in events that can be turned around. For example, if after making the above observations, we now observe Z following Y with a certain interval of time between them, can we assume, based on our observations, that: If they are "temporally related" then X will occur in period Y<->Z * 2 after Y occurs? That is, can we infer Y<->X = (Y<->Z)*2? NOW | <--Past | Future--> | Z Y | X t|---------------------------------------------> \ Prediction (Y<->X)=(Y<->Z)*2 Obviously this is a simplified and idealized picture. In the real world, we may have some "slop" (error constant) in the measured interval differences. Also, three events is the minimum number required for observing the strictly defined ratio-temporality defined here. Three were used here purely for simplicity and clarity. One example of a more useful tactic might be to express the problem/solution in terms of probability. For example, if we've measured a statistically valid sample and observed the temporal ratios to be within +/-10% of their temporally predicted periods, 100% of the time, then we can say our prediction has a reasonable assurance of being accurate to within +/-10% of the predicted time (where assurance is based primarily on the size of the previous sample). If our sample shows that 98% of these events occur within 9% of their temporally predicted times, then we can say that the odds of our predicted time being accurate to within 9% is still reasonably assured, but less assured than our prediction that it will occur within 10% of our prediction. The assurance level, in each case, is based on the sample size, or the number of past observations that came within the stated accuracy range.


. . . Note 1: Are these three components irreducible? It seems as though the three constituent components documented here: frequency, sequence, and temporality are elemental components of (events in) time. It doesn't seem that these three components can themselves be reduced any further, though, to be honest, this is little more than a hunch, as I have not tried very hard to further reduce these components. . . . Note 2: Are there any other components? Intuitively, it seems as though you can also conclude that frequency, sequence, and temporality are the complete set comprising the time domain. In other words, if you convert time to these three domains, you will have enough information to precisely reconstruct the original time domain (within limits defined only by your sample rate). . . . Note 3: Is there any synergy in these components? There may be a way to reconstruct one missing domain given the information from the other two. Can it be done? And is there symmetry (works no matter which two you start with)? . . . Note 4: Representing context Question (speculation): will extra dimensions, such as the addition of a second dimension in the arrow-key example in figure-9C, introduce a possible consistent mechanism for representing context?

Why I Wrote This Paper

I've had a fascination with neural networks and trying to keep up with our ever growing understanding of how biological neural nets function since the 1980's. While experimenting with neural network simulations there has often been a need for a precisely defined understanding of the meaning of temporal and temporality (or whatever it is that has been described here). Clearly this paper takes some liberties. There is some presumption here as to what the word temporal means. The excuse is also the same as the reason for this paper being written. In twenty years of looking I've been unable to find this sub-definition for the word "temporal" or what temporality is. If there is a better fitting word for the concept described here, please let me know. I'd be more than happy to use it. Also, though temporality is not in itself a part of neural network technology, the lack of a labeled understanding for the concept described here (or my inabillity to find one) has been a constant obstacle over the years in various attempts to better understand neural networks. So, with that bit of cautious, tentative, confidence that this wheel just isn't out there, I've decided to invent my own here. Apologies to anyone who, like me, cringes at this. You may find some solace in knowing that this article is not proposing a replacement definition, just a more restricted sense of the word for when context requires. This is important to note. The roadblocks that have emerged needed a definition of the concept of ratio-temporality in GENERAL terms, so this paper seeks to define ratio-temporality in GENERAL terms. How that definition relates to neural networks is outside of the scope of this paper. If you know of anyone out there who has tackled this subject matter already (likely better than I), you will find a grateful student if you could deliver a note stating where to find the material. If you can help, or if you'd just like to discuss it, contact me.


time 1 2 3 4 5 interval 1234567890123456789012345678901234567890123456789012345 event name Q G A X L -Figure 10- Haiku: New Seasons . . . Not the intervals But how they are related Temporality

Reference (historical)

Netlab (A0) version 0.1 April 1990 D. John Repici Netlab v 0.1 is my very early (and first) attempt at a neural network simulation program along with some documentation. The documentation and small test networks include some rudimentary experiments and speculation about how temporal phenomena may play a part in biological neural network learning and operation. This was distributed in April of 1990 on the AI-Expert forum on Compuserve and shortly after on the AI forum on GEnie. Distributions from the AI-Expert forum can still be found on the Fido network.

Sources & Resources


This article is © Copyright, Creativyst, Inc. 2004 - 2013 ALL RIGHTS RESERVED.

Links to this article are always welcome.

However, you may not copy, modify, or distribute this work or any part of it without first obtaining express written permission from Creativyst, Inc. Production and distribution of derivative products, such as displaying this content along with directly related content in a common browser view are expressly forbidden!

Those wishing to obtain permission to distribute copies of this article or derivatives in any form should contact me.

Permissions printed over any code, DTD, or schema files are supported as our permission statement for those constructs.

© Copyright 2004-2013, Dominic John Repici
(only minor edits and references have been added to this online version since 2009)
Written by: Dominic John Repici