Cognitive Algorithm: hierarchical search and pattern encoding

Intelligence is ability to predict and plan (self-predict), which can only be done by discovering and projecting patterns. This definition is well established: pattern recognition is a core of any IQ test. The problem is, there was no general and constructive definition of either pattern or recognition (similarity). So, I came up with my own definitions, which directly translate into algorithm introduced in the next section.

For excellent popular introduction to cognition-as-prediction thesis see “On Intelligence” by Jeff Hawkins and “How to Create a Mind“ by Ray Kurzweil. But on a technical level, both of them and most current researchers  implement pattern discovery via artificial neural networks, which operate in a very coarse fashion:

ANN learns via some version of Hebbian “fire together, wire together” coincidence reinforcement. Normally, “neuron’s” inputs are weighed at synapses, then summed and thresholded into output. Output also back- propagates to synapses and is combined with their last input to adjust the weights. This weight adjustment is learning. But prior summation degrades initial resolution of inputs, precluding any comparison between them (the inputs are recoverable, but why degrade and then spend a ton of computation restoring them).

I believe resolution of inputs should be adjusted according to their predictive value, not reduced by default. Predictive value is input’s match to future inputs, projected from its match to past inputs. I define match as lossless compression per input (part 1, compare to existing measures of similarity), from cross-comparing inputs at initial resolution. Then inputs should be cross-compared over extended distance within patterns: spans of above-average match, and summed within spans of both above- and below- average match.

Statistical methods work in reverse order: inputs are summed within arbitrary samples, then the sums are evaluated as feedback to redefine weights and future samples into some meaningful shape. Neural Nets are all the rage now, and they obviously work, especially in a deep learning hierarchy. But they require very long training, thus don’t scale without supervision or task-specific reinforcement, and also recognize things that aren’t there. These problems are caused by aforementioned coarseness, inherent in statistical learning.

Currently the most successful method is CNN, which defines match as a product: input * kernel (weights). Which does overweigh similar vs. distant input-kernel pairs, relative to their sum. But such similarity is exaggerated: match is a common subset of comparands, not their superset. Exaggerated match would distort prediction by underweighing competing co-derived difference: input - kernel. And ANN don’t even compute such difference. I think that’s what causes initialization bias in ANN and similar confirmation bias in humans.

In other words, weights (kernels) are not directly adjusted by their cumulative difference to past inputs, decoupling training from inference. My equivalent of inference is comparison, which forms separate partial match and difference. Feedback of this difference updates (trains) my filters: less specific version of weights.
Also, both input and kernel are arrays. This is far more coarse, thus less selective and efficient, than one-to-one comparison of adjacent inputs in my algorithm (initial inputs are pixels, higher-level inputs are their patterns).

Inspiration by the brain kept ANN research going for decades before they became useful. But their “neurons” are mere stick figures for real ones. Of course, most of complexity in a neuron is due to constraints of biology. Ironically, weighted summation in ANN may also be a no-longer needed compensation for such constraint:
neural memory requires dedicated connections (synapses), which makes individual input representation and comparison prohibitively expensive. But not anymore, we now have dirt-cheap random access memory.

Other biological constraints are very slow neurons, and the imperative of fast reaction for survival in the wild. Both favor fast though crude summation (vs. slower one-to-one comparison), at the cost of glacial training. Reaction speed became less important: modern society is quite secure, while continuous learning is far more important because of accelerating progress. Another constraint is noise: neurons often fire at random, so their spikes are summed to get a stable input. But that’s not a reason to degrade far more precise electronic signals.   

Current Machine Learning and related theories (AIT, Bayesian inference, etc.) are largely statistical also because they were developed primarily for symbolic data. Such data, pre-compressed and pre-selected by humans, is far more valuable than sensory inputs it was ultimately derived from. But due to this selection and compression, proximate symbols are not likely to match, and partial match between them is hard to quantify. Hence, symbolic data is a misleading initial target for developing conceptually consistent algorithm.

outline of my approach

Proposed algorithm is a clean design for deep learning: non-neuromorphic, sub-statistical, comparison-first.
It’s a hierarchical search for patterns, similar to hierarchical clustering. But conventional clustering defines match as inverted difference between inputs, which is wrong. Match is a subset common for both comparands, distinct from and complementary to their difference. Which is a type of miss, to be formed along with match.

I quantify match and miss by cross-comparing inputs over selectively extended range of search. Basic comparison is inverse arithmetic operation between two single-variable inputs, starting with adjacent pixels.
Specific match and miss is determined by power of comparison: Boolean match is AND and miss is XOR, comparison by subtraction increases match to a smaller comparand and reduces miss to a difference,
comparison by division increases match to a multiple and reduces miss to a fraction, and so on (part 1).

My pattern is a span of matching input patterns, and match between patterns is a combined match between their variables. Search expansion should be strictly incremental, to allow for fine-grained input selection. Within each level, search is incremental in distance between inputs and in their derivation (part 4, level 1). Between levels, it is incremental in compositional scope and number of derived variables per pattern. “Strictly incremental” means there is a unique set of operations per level, hence a singular in “cognitive algorithm“.

First level has single-variable inputs, such as pixels in video. Then each level derives new variables: match and miss, per every compared variable of an input pattern. So, the variables multiply at every level, new variables are summed within a pattern, then combined to evaluate it for expanded search. Feedback of new variables extends lower-level algorithm by adding operations to filter future inputs, and then by adjusting these filters.
I don’t know of any clustering that implements such incremental syntax: number of variables per pattern.

Actually, recently introduced “capsules” also output multivariate vectors, similar to my patterns. But their core input is a probability estimate from unrelated method: CNN, while all variables in my patterns are derived by incrementally complex comparison. In a truly general method, the same principles must apply on all stages of processing. And additional variables in capsules are only positional, while my patterns also add differences between input variables. That can’t be done in capsules because differences are not computed by CNN.

Autonomous cognition must start with analog inputs, such as video or audio. All symbolic data, including natural languages, is encoded by some cognitive process. Thus, it should be decoded before being cross- compared to search for meaningful patterns. And the difficulty of decoding is exponential with the level of encoding, so hierarchical learning starting with raw sensory input is by far the easiest to implement (part i).
Hence, my initial inputs are pixels, and higher-level inputs are patterns formed by lower-level search.

To discover anything complex at “polynomial” cost, resulting patterns should also be hierarchical: each level of search adds a level of composition and sub-differentiaton of its input patterns. Higher-level search should be selective per resulting level of pattern. Composition and selection reduces search, which forms longer-range spatio-temporal and then conceptual patterns. They also send feedback: filters and then motor action, to select lower-level inputs and locations with above-average additive predictive value (part 3).

Hierarchical approaches are common in unsupervised learning, and all do some sort of pattern recognition.
But none that I know of is strictly incremental in scope and complexity of discoverable patterns. Incremental  selection is necessary for scalability, to avoid combinatorial explosion in search space. But it’s more expensive upfront and won’t pay in simple test problems. So, it’s not suitable for immediate experimentation, which is probably why no one else seems to be working on anything sufficiently similar to my algorithm.

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elaboration below:

i.  Cognition vs. evolution, analog vs. symbolic initial input
1.  Comparison: quantifying match and miss between two variables
2. Forward search and patterns, incremental space-time dimensionality
3. Feedback of filters, attentional input selection, imagination, motor action

4. Initial levels of search, corresponding orders of feedback, and resulting patterns:

level 1: comparison to past inputs, forming difference and relative match patterns
level 2: additional evaluation of resulting patterns for feedback, forming filter patterns
level 3: additional evaluation of projected filter patterns, forming updated-input patterns

5. Comparison between variable types within a pattern
6. Cartesian dimensions and sensory modalities
7. Notes on clustering, ANNs, and probabilistic inference
8. Notes on working mindset, “hiring” and a prize for contributions

i. Cognition vs. evolution, analog vs. symbolic initial input

Some say intelligence can be recognized but not defined. I think that’s absurd: we recognize some implicit definition. Others define intelligence as a problem-solving ability, but the only general problem is efficient search for solutions. Efficiency is a function of selection among inputs, vs. brute-force all-to-all search. This selection is by predicted value of the inputs, and prediction is interactive projection of their patterns. Some agree that intelligence is all about pattern discovery, but define pattern as a crude statistical coincidence.

Of course, the only mechanism known to produce human-level intelligence is even cruder, and that shows in haphazard construction of our brains. Algorithmically simple, biological evolution alters heritable traits at random and selects those with above-average reproductive fitness. But this process requires almost inconceivable computing power because selection is very coarse: on the level of whole genome rather than individual traits, and also because intelligence is only one of many factors in reproductive fitness.

Random variation in evolutionary algorithms, generative RBMs, and so on, is antithetical to intelligence. Intelligent variation must be driven by feedback within cognitive hierarchy: higher levels are presumably “smarter” than lower ones. That is, higher-level inputs represent operations that formed them, and are evaluated to alter future lower-level operations. Basic operations are comparison and summation among inputs, defined by their range and resolution, analogous to reproduction in genetic algorithms.

Range of comparison per conserved-resolution input should increase if projected match (cognitive fitness function) exceeds average match per comparison. In any non-random environment, average match declines with the distance between comparands. Thus, search over increasing distance requires selection of above- average comparands. Any delay, coarseness, and inaccuracy of such selection is multiplied at each search expansion, soon resulting in combinatorial explosion of unproductive (low additive match) comparisons.

Hence, my model is strictly incremental: search starts with minimal-complexity inputs and expands with minimal increments in their range and complexity (syntax). At each level, there is only one best increment, projected to discover the greatest additive match. No other AGI approach follows this principle.
I guess people who aim for human-level intelligence are impatient with small increments and simple sensory data. Yet, this is the most theoretical problem ever, demanding the longest delay in gratification.

symbolic obsession and its discontents

Most obvious example of this impatience is the use of symbolic data as initial inputs in cognitive projects. Such a start betrays profound misunderstanding of cognition. Even children, predisposed to learn language, only become fluent after years of directly observing things their parents talk about. Words are mere labels for concepts, the most important of which are generalized spatio-temporal patterns, derived from multi-modal sensory experience. Top-down reconstruction of such patterns solely from correlations among their labels is exponentially more difficult than their bottom-up construction from sensory inputs.    

All our knowledge is ultimately derived from senses, but lower levels of human perception are unconscious. Only generalized concepts make it into our consciousness, AKA declarative memory, where we assign them symbols (words) to facilitate communication. This brain-specific constraint creates heavy symbolic vs. sub-symbolic bias, especially strong in artificial intelligentsia. Which is like putting a cart in front of a horse: most words are meaningless unless coupled with representations of sensory patterns.

To be incrementally selective, cognitive algorithm must exploit proximity first, which is only productive for continuous  and loss-tolerant raw sensory data. Symbolic data is already compressed: consecutive characters and words in text won’t match. It’s also encoded with distant cross-references, that are hardly ever explicit outside of a brain. Text looks quite random unless you know the code: operations that generalized pixels into patterns (objects, processes, concepts). That means an algorithm designed specifically for text will not be consistently incremental in the range of search, which will impair its scalability.

In Machine Learning, input is string, frame, or video sequence of a defined length, with artificial separation between training and inference. In my approach, learning is continuous and interactive. Initial inputs are streamed pixels of maximal resolution, and higher-level inputs are multi-variate patterns formed by comparing lower-level inputs. Spatio-temporal range of inputs, and selective search across them, is extended indefinitely. This expansion is directed by higher-level feedback, just as it is in human learning.

Everything ever written is related to my subject, but nothing is close enough: not other method is meant to be fully consistent. Hence a dire scarcity of references here. My approach is presented bottom-up (parts 1 - 6), thus can be understood without references. But that requires a clean context, - hopefully cleaned out by reader‘s own introspective generalization. Other (insufficiently) related approaches are addressed above and in part 7. I also have a more advanced work-in-progress, but will need a meaningful feedback to elaborate.

1. Comparison: quantifying match and miss between two variables

First of all, we must quantify predictive value. Algorithmic information theory defines it as compressibility of representation, which is perfectly fine. But current implementations only define compression for whole sequences of inputs. To enable far more incremental selection (thus more scalable search), I start by quantifying match (= compression) between individual inputs. Partial match is a finer dimension of analysis, vs. binary same | different inputs in probabilistic inference. This is similar to the way probabilistic inference improved on classical logic, by quantifying partial probability vs. binary true | false value of statements.

I define match as a complementary of miss. Basic miss is a difference between comparands, hence match is a smaller comparand. In other words, match is a compression of larger comparand’s magnitude by replacing it with difference relative to smaller comparand. Ultimate criterion is recorded magnitude, rather than bits of memory it occupies, because the former represents physical impact that we want to predict. The volume of memory used to record that magnitude depends on prior compression, which is not an objective parameter.

This definition is tautological: smaller input is a common subset of both inputs, = sum of AND between their uncompressed (unary code) representations. Some may object that match includes the case when both inputs equal zero, but that match also equals zero. The purpose is prediction, which is a representational equivalent of conservation in physics. Ultimately, we’re predicting some potential impact on observer, represented by input. Zero input means zero impact, which has no conservable inertia, thus no intrinsic predictive value.

Given incremental complexity of representation, initial inputs should have binary resolution. However, these binary inputs are more efficiently compressed by digitization within a coordinate: substitution of every two lower-order bits of an integer with one higher-order bit. Resolution of coordinate (input summation span) is adjusted to form large enough integers, to produce average match that exceeds the cost of comparison.
All costs here are opportunity costs: average compression by equivalent computational resources.

Next order of compression is by comparing integers across coordinates, initially only between consecutive coordinates. Basic comparison is inverse arithmetic operation of incremental power: AND, subtraction, division, logarithm, and so on. Additive match is achieved by comparison of a higher power than that which produced comparands: comparison by AND will not further compress integers previously digitized by AND.

Rather, initial comparison between integers is by subtraction, miss is resulting difference and absolute match is a smaller input. For example, if inputs are 4 and 7, then miss is 3, and their match or common subset is 4. Difference is smaller than XOR (complementary of AND) because XOR includes opposite-sign bit pairs 0,1 and 1,0, which will be cancelled-out by subtraction.

Comparison by division forms ratio, which is a compressed difference. Compression is explicit in long division: match is accumulated over iterative subtraction of smaller comparand from remaining difference. In other words, this is also a comparison by subtraction, but between different orders of derivation. New match is smaller comparand * integer part of ratio, and miss is final reminder or fractional part of ratio.

Ratio can be further compressed by converting it to radix or logarithm, and so on.
By reducing miss, higher-power comparison increases complementary match (match = larger input - miss):

total per order:   1) larger input   2) XOR                       3) difference: combined current-order match & miss
additive match:       AND                opposite-sign XOR      multiple of a smaller input within a difference
remaining miss:      XOR                difference                       fraction: complementary to multiple within a ratio

But the costs of operations and record of signs, fractions, irrational fractions, etc. may grow even faster. Thus, the power of comparison is increased only if higher-level “meta-comparison” between results from different powers of comparison finds a pattern of improving (compression - costs) with increasing power.

The above covers absolute match, which must be projected on potential search location before evaluation. Projection is recombination of co-derived matches and misses at a distant location:
projected match = match + ((difference * (summed match / summed input)) * distance) / 2  

This match projection from results of a single comparison is not implemented in the algorithm. That’s because projection is a feedback, which adjusts lower-level filters and is represented in the output. Incurred costs must be justified by the value of feedback, which is low per comparison but accumulates over multiple comparisons.

So, actual feedback is *average* projected match in higher-level patterns, formed by multiple vs. individual comparisons. This projection is by recombining overlapping match patterns and difference patterns in proportion to their predictive value (vs. recombining individual matches and differences above).

But ultimate selection criterion is relative match: current-level match - past current-level match that co-occurs with average higher-level projected match. In other words, relative match is positive or negative predictive value, which determines input inclusion into positive or negative pattern.

2. Forward search and patterns, incremental space-time dimensionality

Pattern is a contiguous span of inputs that form above-average matches, similar to conventional cluster.
As explained above, matches and misses (derivatives) are produced by comparing consecutive inputs. These derivatives are summed within a pattern and then compared between patterns on the next level of search, adding new derivatives to a higher pattern. Patterns are defined contiguously for each level, but are necessarily followed by a span of below-average match (gap or negative pattern), thus next-level inputs are discontinuous.

Negative patterns represent contrast or discontinuity between positive patterns, which is a one- or higher- dimensional equivalent of difference between zero-dimensional pixels. As with differences, projection of a negative pattern competes with projection of adjacent positive pattern. But match and difference is derived from the same input pair, while positive and negative patterns represent separate spans of inputs.

Negative pattern is not predictive on its own, but is valuable for allocation: computational resources of no-longer predictive pattern should be used elsewhere. Hence, the value of negative pattern is borrowed from predictive value of co-projected positive pattern, as long as combined additive match remains above average.
Consecutive positive and negative patterns project over same future input span, and these projections partly cancel each other. So, they should be combined to form feedback, as explained in part 3.

Initial match is evaluated for inclusion into higher positive or negative pattern. The value is summed until its sign changes, and if positive, evaluated again for cross-comparison among constituent inputs over increased distance. Second evaluation is necessary because the cost of incremental syntax, produced by cross-comparing, is per pattern rather than per input. Pattern is terminated and outputted to the next level when value sign changes. On the next level, it is compared to previous patterns of the same compositional order.

Initial inputs are pixels of video, or equivalent limit of positional resolution in other modalities. Hierarchical search on higher levels should discover patterns representing empirical objects and processes, and then relational logical and mathematical shortcuts, eventually exceeding generality of our semantic concepts. Which are also patterns in cognitive terms: all information is either learned as patterns or filtered out as noise. For online learning, all levels should receive inputs from lower levels and feedback from higher levels in parallel.

space-time dimensionality and initial implementation

Any prediction has two components: what and where. We must have both: value of prediction = precision of what * precision of where. That “where” is currently neglected: statistical ML represents space-time at greatly reduced resolution, if at all. In the brain and some neuromorphic models, “where” is represented in a separate network. That makes transfer of locations very expensive and coarse, reducing predictive value of combined representations. There is no such immediate degradation of positional information in my method.

My core algorithm is 1D: time only (part 4). Our space-time is 4D, but each of  these dimensions can be mapped on one level of search. This way, levels can select input patterns that are strong enough to justify the cost of representing additional dimension, as well as derivatives (matches and differences) in that dimension.
Initial 4D cycle of search would compare contiguous inputs, similarly to connected-component analysis:

level 1 compares consecutive 0D pixels within horizontal scan line, forming 1D patterns: line segments.
level 2 compares contiguous 1D patterns between consecutive lines in a frame, forming 2D patterns: blobs.
level 3 compares contiguous 2D patterns between incremental-depth frames, forming 3D patterns: objects.
level 4 compares contiguous 3D patterns in temporal sequence, forming 4D patterns: processes.

Subsequent cycles would compare 4D input patterns over increasing distance in each dimension, forming longer-range discontinuous patterns. These cycles can be coded as implementation shortcut, or discovered by core algorithm itself, which can adapt to inputs of any dimensionality. “Dimension” here is a parameter that defines external sequence and distance among inputs. This is different from conventional clustering, which treats both external and internal parameters as dimensions. More in part 6.

However, average match in our space-time is presumably equal over all four dimensions. That means patterns defined in fewer dimensions will be biased by the angle of scanning, introducing artifacts. Hence, initial pixel comparison and inclusion into patterns should also be defined over 4Ds, or at least over 2D blobs for still images. This is a universe-specific extension of my core algorithm.
I am currently working on implementation of core algorithm adapted to process images and then video, but it should be extensible to any type and scope of data. My work-in-progress Python code is posted on github, suggestions and collaboration are most welcome, see the last part here on prizes.

3. Feedback of filters, attentional input selection, imagination, motor action

(needs work)

After evaluation for inclusion into higher-level pattern, input is also evaluated as feedback to lower levels. Feedback is update to filters that evaluate forward (Λ) and feedback (V), as described above but on lower level.
Basic filter is average value of input’s projected match that co-occurs with (thus predicts) average higher-level match within a positive (above-average) pattern. Both values are represented in resulting patterns.

Feedback value = forward value - value of derivatives /2, both of an input pattern. In turn, forward value is determined by higher-level feedback, and so on. Thus, all higher levels affect selection on lower-level inputs. This is because the span of each pattern approximates, hence projects over, combined span of all lower levels.
Indirect feedback propagates level-sequentially, more expensive shortcut feedback is sent to selected levels.

Negative derivatives project increasing match: match to subsequent inputs is greater than to previous inputs.
Such feedback will reduce lower-level filter. If filter is zero, all inputs are cross-compared, and if filter is negative, it is applied to cancel subsequent filters for incrementally longer-range cross-comparison.
There is one filter for each compared variable within input pattern, initialized at 0 and updated by feedback.

novelty vs. generality

Any integrated system must have a common selection criterion. Two obvious cognitive criteria are novelty and generality: miss and match, But we can’t select for both, they exhaust all possibilities. Novelty can’t be primary criterion: it would select for noise and filter out all patterns, which are defined by match. On the other hand, to maximize match of inputs to memory we can “stare at a wall” and lock into predictable environments. But of course, natural curiosity actively skips predictable locations, thus reducing the match.

This dilemma is resolved if we maximize predictive power: projected match, rather than actual match, of inputs to records. To the extent that new match was projected by past inputs, it doesn’t add to their projected match. But neither does noise: novelty (difference to past inputs) that is not projected to persist (match) in the future.
Additive projected match = downward (V) match to subsequent inputs - upward (Λ) match to previous inputs.

This ΛV asymmetry comes from projecting difference or value (both are signed) of input relative to feedback.
Λ match is initially projected from match between input and its higher-level average. Such averages may be included in feedback, along with average match (filter). Each average is compared to same-type variable of an input, and resulting match (redundancy to a higher level) is subtracted from input match before evaluation.

Averages represent past vs. current higher level and should be projected over feedback delay:
average += average difference * (delay / average span) /2. Value of input-to-average match is reduced by higher-level match rate: rM= match / input, so additive match = input match - input-to-average match * rM.
Adjustment by input-to-average match is done if rM is above-average for incurred cost of processing.

So, basic selection for novelty is subtraction of adjusted Λ match-to-filter. V selection for generality, on the other hand, is a function of back-projected difference of match, which is a higher-derivation feedback. Projected increase of match adds to, and decrease of match subtracts from, predictive value of future inputs.
Higher-order selection for novelty should skip (or avoid processing) future input spans predicted with high certainty. This is a selection of projected vs. actual inputs, covered in part 4, level 3.  

More generally, ΛV projection asymmetry is expressed by differences of incremental representation order:
difference in magnitude of initial inputs: projected next input = last input + difference/2,
difference in input match, a subset of magnitude: projected next match = last match + match difference/2,
difference in match of match, a sub-subset of magnitude, projected correspondingly, and so on.
Ultimate criterion is top order of match on a top level of search: the most predictive parameter in a system.

imagination, planning, action

Imagination is not actually original, it can only be formalized as interactive projection of known patterns. Strong patterns send positive feedback to lower-level locations where they are projected to re-occur at higher resolution, to increase represented detail. When originally distant patterns are projected to re-occur in the same location, their projections interact and combine. This is a generative (vs. reductive) mechanism.

Interaction is comparison and summation between same-type variables of co-projected patterns: direct feedback from multiple higher levels to the same location. Location is a span of search within a lower level, which receives feedback with matching projected coordinates. Evaluation of actual inputs is delayed until additive match projected in their location exceeds the value of combined filters.

Comparison forms patterns of filters to compress their representations and selectively project their match, same as with original inputs. Summation of their differences cancels or reinforces corresponding variables, forming combined filter, including secondary patterns of repulsion or attraction. These patterns per variable are discovered on a higher level, by comparing past inputs with their multiple-feedback projections.

Combined filter is pre-evaluated: projected value of positive patterns is compared to projected cost of evaluating all inputs, both within a filtered location. Resulting prevalue (value of evaluation) is negative when the latter exceeds the former: projected inputs are not worth evaluating and their span / location is skipped.

Iterative skipping of input spans is the most basic motor feedback: it increments coordinates and differences of the filters. Cross-filter search then proceeds over multiple “imagined” locations, before finding one with projected above-average additive match. That’s where skipping stops and actual inputs are received.

Cognitive component of action is planning. It is technically identical to imagination, except that projected patterns also represent the system itself. Feedback of such self-patterns eventually reaches the bottom of representational hierarchy: sensors and actuators, adjusting their sensitivity and coordinates. Such environmental interface is part of any cognitive system, although actuators are optional.

4. Initial levels of search, corresponding orders of feedback and resulting patterns

This part recapitulates and expands on my core algorithm, which operates in one dimension: time only. Spatial and derived dimensions are covered in part 6. Even within 1D, search is hierarchical in scope, containing any number of levels. New level is added when current top level terminates and outputs the pattern it formed.

Higher-level patterns are fed back to select future inputs on lower levels. Feedback is sent to all lower levels because span of each pattern approximates combined span of inputs within whole hierarchy below it.
So, deeper hierarchy forms higher orders of feedback, with increasing elevation and scope relative to its target: same-level prior input, higher-level match average, beyond-the-next-level match value average, etc.

These orders of feedback represent corresponding order of input compression: input, match between inputs, match between matches, etc. Such compression is produced by comparing inputs to feedback of all orders. Comparisons form patterns, the type of which corresponds to relative span of compared feedback:

1: prior inputs are compared to the following ones on the same level, forming difference patterns dPs,
2: feedback of higher-level match is used to evaluate match between inputs, forming value patterns vPs,
3: trans-level feedback re-evaluates positive values of match, forming more selective shortcut patterns sPs

Feedback of 2nd order consists of input filters (if) defining value patterns, and coordinate filters (Cf) defining positional resolution and relative distance to future inputs.
Feedback of 3rd order is shortcut filters for beyond-the-next level. These filters, sent to a location defined by attached coordinate filters, form higher-order value patterns for deeper internal and distant-level comparison.

Higher-order patterns are more selective: difference is as likely to be positive as negative, while value is far more likely to be negative, because positive patterns add costs of re-evaluation for extended cross-comparison among their inputs. And so on, with selection and re-evaluation for each higher order of positive patterns. Negative patterns are still compared as a whole: their weak match is compensated by greater span.

All orders of patterns formed on the same level are redundant representations of the same inputs. Patterns contain representation of match between their inputs, which are compared by higher-order operations. Such operations increase overall match by combining results of lower-order comparisons across pattern’s variables:

0Le: AND of bit inputs to form digitized integers, containing multiple powers of two
1Le: SUB  of integers to form patterns, over additional external dimensions = pattern length L
2Le: DIV  of multiples (L) to form ratio patterns, over additional distances = negative pattern length LL
3Le: LOG of powers (LLs), etc.  Starting from second level, comparison is selective per element of an input.

Such power increase also applies in comparison to higher-order feedback, with a lag of one level per order.
Power of coordinate filters also lags the power of input filters by one level:
1Le fb: binary sensor resolution: minimal and maximal detectable input value and coordinate increments   
2Le fb: integer-valued average match and relative initial coordinate (skipping intermediate coordinates)
3Le fb: rational-valued coefficient per variable and multiple skipped coordinate range
4Le fb: real-valued coefficients and multiple coordinate-range skip

I am defining initial levels to find recurring increments in operations per level, which could then be applied to generate higher levels recursively, by incrementing syntax of output patterns and of feedback filters per level.

operations per generic level  (out of date)

Level 0 digitizes inputs, filtered by minimal detectable magnitude: least significant bit (i LSB). These bits are AND- compared, then their matches are AND- compared again, and so on, forming integer outputs. This is identical to iterative summation and bit-filtering by sequentially doubled i LSB.

Level 1 compares consecutive integers, forming ± difference patterns (dP s). dP s are then evaluated to cross-compare their individual differences, and so on, selectively increasing derivation of patterns.
Evaluation: dP M (summed match) - dP aM (dP M per average match between differences in level 2 inputs).

Integers are limited by the number of digits (#b), and input span: least significant bit of coordinate (C LSB).
No 1st level feedback: fL cost is additive to dP cost, thus must be justified by the value of dP (and coincident difference in value of patterns filtered by adjusted i LSB), which is not known till dP is outputted to 2nd level.

Level 2 evaluates match within dP s | bf L (dP) s, forming ± value patterns: vP s | vP (bf L) s. +vP s are evaluated for cross-comparison of their dP s, then of resulting derivatives, then of inputted derivation levels. +vP (bf L) s are evaluated to cross-compare bf L s, then dP s, adjusted by the difference between their bit filters, and so on.

dP variables are compared by subtraction, then resulting matches are combined with dP M (match within dP) to evaluate these variables for cross-comparison by division, to normalize for the difference in their span.
// match filter is also normalized by span ratio before evaluation, same-power evaluation and comparison?

Feedback: input dP s | bf L (dP) are back-projected and resulting magnitude is evaluated to increment or decrement 0th level i LSB. Such increments terminate bit-filter span ( bf L (dP)), output it to 2nd level, and initiate a new i LSB span to filter future inputs. // bf L (dP) representation: bf , #dP, Σ dP, Q (dP).

Level 3 evaluates match in input vP s or f L (vP) s, forming ± evaluation-value patterns: eP s | eP (fL) s. Positive eP s are evaluated for cross-comparison of their vP s ( dP s ( derivatives ( derivation levels ( lower search-level sources: buffered or external locations (selected sources may directly specify strong 3rd level sub-patterns).

Feedback: input vP is back-projected, resulting match is compared to 2nd level filter, and the difference is evaluated vs. filter-update filter. If update value is positive, the difference is added to 2nd level filter, and filter span is terminated. Same for adjustment of previously covered bit filters and 2nd level filter-update filters?

This is similar to 2nd level operations, but input vP s are separated by skipped-input spans. These spans are a filter of coordinate (Cf, higher-order than f for 2nd level inputs), produced by pre-valuation of future inputs:
projected novel match = projected magnitude * average match per magnitude - projected-input match?

Pre-value is then evaluated vs. 3rd level evaluation filter + lower-level processing cost, and negative prevalue-value input span (= span of back-projecting input) is skipped: its inputs are not processed on lower levels.
// no prevaluation on 2nd level: the cost is higher than potential savings of only 1st level processing costs?

As distinct from input filters, Cf is defined individually rather than per filter span. This is because the cost of Cf update: span representation and interruption of processing on all lower levels, is minor compared to the value of represented contents? ±eP = ±Cf: individual skip evaluation, no flushing?

or interruption is predetermined, as with Cb, fixed C f within C f L: a span of sampling across fixed-L gaps?
alternating signed Cf s are averaged ±vP s?
Division: between L s, also inputs within minimal-depth continuous d-sign or m-order derivation hierarchy?

tentative generalizations and extrapolations

So, filter resolution is increased per level, first for i filters and then for C filters: level 0 has input bit filter,
level 1 adds coordinate bit filter, level 2 adds input integer filter, level 3 adds coordinate integer filter.
// coordinate filters (Cb, Cf) are not input-specific, patterns are formed by comparing their contents.

Level 4 adds input multiple filter: eP match and its derivatives, applied in parallel to corresponding variables of input pattern. Variable-values are multiplied and evaluated to form pattern-value, for inclusion into next-level ±pattern // if separately evaluated, input-variable value = deviation from average: sign-reversed match?

Level 5 adds coordinate multiple filter: a sequence of skipped-input spans by iteratively projected patterns, as described in imagination section of part 3. Alternatively, negative coordinate filters implement cross-level shortcuts, described in level 3 sub-part, which select for projected match-associated novelty.

Additional variables in positive patterns increase cost, which decreases positive vs. negative span proportion.
Increased difference in sign, syntax, span, etc., also reduces match between positive and negative patterns. So, comparison, evaluation, pre-valuation... on higher levels is primarily for same-sign patterns.

Consecutive different-sign patterns are compared due to their proximity, forming ratios of their span and other variables. These ratios are applied to project match across different-sign gap or contrast pattern:
projected match += (projected match - intervening negative match) * (negative value / positive value) / 2?

ΛV selection is incremented by induction: forward and feedback of actual inputs, or by deduction: algebraic compression of input syntax, to find computational shortcuts. Deduction is faster, but actual inputs also carry  empirical information. Relative value of additive information vs. computational shortcuts is set by feedback.

Following sub-parts cover three initial levels of search in more detail, though out of date:

Level 1: comparison to past inputs, forming difference patterns and match patterns

Inputs to the 1st level of search are single integers, representing pixels of 1D scan line across an image, or equivalents from other modalities. Consecutive inputs are compared to form differences, difference patterns, matches, relative match patterns. This comparison may be extended, forming higher and distant derivatives:

resulting variables per input: *=2 derivatives (d,m) per comp, + conditional *=2 (xd, xi) per extended comp:

     8 derivatives   // ddd, mdd, dd_i, md_i, + 1-input-distant dxd, mxd, + 2-input-distant d_ii, m_ii,
           /        \
      4 der   4 der     // 2 consecutive: dd, md, + 2 derivatives between 1-input-distant inputs: d_i and m_i,
      /     \    /    \
   d,m   d,m   d,m    // d, m: derivatives from default comparison between consecutive inputs,
  /    \   /    \  /     \
i  >>  i  >>  i  >>  i    // i: single-variable inputs.

This is explained / implemented in my draft python code:  level_1_working. That first level is for generic 1D cognitive algorithm,  its adaptation for image and then video recognition algorithm will be natively 2D.
That’s what I spend most of my time on, the rest of this intro is significantly out of date.

bit-filtering and digitization   

1st level inputs are filtered by the value of most and least significant bits: maximal and minimal detectable magnitude of inputs. Maximum is a magnitude that co-occurs with average 1st level match, projected by outputted dP s. Least significant bit value is determined by maximal value and number of bits per variable.

This bit filter is initially adjusted by overflow in 1st level inputs, or by a set number of consecutive overflows.
It’s also adjusted by feedback of higher-level patterns, if they project over- or under- flow of 1st level inputs that exceeds the cost of adjustment. Underflow is average number of 0 bits above top 1 bit.
Original input resolution may be increased by projecting analog magnification, by impact or by distance.

Iterative bit-filtering is digitization: bit is doubled per higher digit, and exceeding summed input is transferred to next digit. A digit can be larger than binary if the cost of such filtering requires larger carry.
Digitization is the most basic way of compressing inputs, followed by comparison between resulting integers.

hypothetical: comparable magnitude filter, to form minimal-magnitude patterns

Initial magnitude justifies basic comparison, and summation of below-average inputs only compensates for their lower magnitude, not for the cost of conversion. Conversion involves higher-power comparison, which must be justified by higher order of match, to be discovered on higher levels. Or by average neg. mag. span?

iP min mag span conversion cost and comparison match would be on 2nd level, but it’s not justified by 1st level match, unlike D span conversion cost and comparison match, so it is effectively the 1st level of comparison?
possible +iP span evaluation: double evaluation + span representation cost < additional lower-bits match?

The inputs may be normalized by subtracting feedback of average magnitude, forming ± deviation, then by dividing it by next+1 level feedback, forming a multiple of average absolute deviation, and so on. Additive value of input is a combination of all deviation orders, starting with 0th or absolute magnitude.

Initial input evaluation if any filter: cost < gain: projected negative-value (comparison cost - positive value):
by minimal magnitude > ± relative magnitude patterns (iP s), and + iP s are evaluated or cross-compared?
or by average magnitude > ± deviations, then by co-average deviation: ultimate bit filter?

Summation *may* compensate for conversion if its span is greater than average per magnitude spectrum?!
Summation on higher levels also increases span order, but within-order conversion is the same, and  between-order comparison is intra-pattern only. bf spans overlap vP span, -> filter conversion costs?

Level 2: additional evaluation of input patterns for feedback, forming filter patterns

(out of date).

Inputs to 2nd level of search are patterns derived on 1st level: dP ( L, I, D, V, Q (d)) s or t dP ( tV, tD, dP ()) s.
These inputs are evaluated for feedback to update 0th level i LSB, terminating same-filter span.
Feedback increment of LSB is evaluated by deviation () of magnitude, to avoid input overflow or underflow:

+= I/ L - LSB a; |∆| > ff? while (|∆| > LSB a){ LSB ±; |∆| -= LSB a; LSB a *2};
LSB a is average input (* V/ L?) per LSB value, and ff is average deviation per positive-value increment;
Σ () before evaluation: no V patterns? #b++ and C LSB-- are more expensive, evaluated on 3rd level?

They are also compared to previously inputted patterns, forming difference patterns dPs and value patterns vPs per input variable, then combined into dPP s and vPP s per input pattern.

L * sign of consecutive dP s is a known miss, and match of dP variables is correlated by common derivation.
Hence, projected match of other +dP and -dP variables = amk * (1 - L / dP). On the other hand, same-sign dP s are distant by L, reducing projected match by amk * L, which is equal to reduction by miss of L?

So, dP evaluation is for two comparisons of equal value: cross-sign, then cross- L same-sign (1 dP evaluation is blocked by feedback of discovered or defined alternating sign and co-variable match projection).
Both of last dP s will be compared to the next one, thus past match per dP (dP M) is summed for three dP s:

dP M ( Σ ( last 3 dP s L+M)) - a dP M (average of 4Le +vP dP M) -> v, vs;; evaluation / 3 dP s -> value, sign / 1 dP.
while (vs = ovs){ ovs = vs; V+=v; vL++; vP (L, I, M, D) += dP (L, I, M, D);; default vP - wide sum, select preserv.

vs > 0? comp (3 dP s){ DIV (L, I, M, D) -> N, ( n, f, m, d); vP (N, F, M, D) += n, f, m, d;; sum: der / variable, n / input?  
vr = v+ N? SUB (nf) -> nf m; vd = vr+ nf m, vds = vd - a;; ratios are too small for DIV?

while (vds = ovds){ ovds = vds; Vd+=vd; vdL++; vdP() += Q (d | ddP);; default Q (d | ddP) sum., select. preserv.
vds > 0? comp (1st x lst d | ddP s of Q (d) s);; splicing Q (d) s of matching dP s, cont. only: no comp ( Σ Q (d | ddP)?

Σ vP ( Σ vd P eval: primary for -P, redundant to individual dP s ( d s  for +P, cost *2, same for +P' I and -P' M,D?
no Σ V | Vd evaluation of cont. comp per variable or division: cost + vL = comp cost? Σ V per fb: no vL, #comp;

- L, I, M, D: same value per mag, power / compression, but I | M, D redund = mag, +vP: I - 2a, - vP: M, D - 2a?
- no variable eval: cost (sub + vL + filter) > comp cost, but match value must be adjusted for redundancy?
- normalization for comparison: min (I, M, D) * rL, SUB (I, M, D)? Σ L (pat) vs C: more general but interrupted?

variable-length DIV: while (i > a){ while (i> m){ SUB (i, m) -> d; n++; i=d;}; m/=2; t=m;  SUB (d, t); f+= d;}?
additive compression per d vs. m*d: > length cost?
tdP ( tM, tD, dP(), ddP Σ ( dMΣ (Q (dM)), dDΣ (Q (dD)), ddLΣ (Q (ddL)), Q (ddP))); // last d and D are within dP()?

Input filter is a higher-level average, while filter update is accumulated over multiple higher-level spans until it exceeds filter-update filter. So, filter update is 2nd order feedback relative to filter, as is filter relative to match.
But the same filter update is 3rd  order of feedback when used to evaluate input value for inclusion into pattern defined by a previous filter: update span is two orders higher than value span.

Higher-level comparison between patterns formed by different filters is mediated, vs. immediate continuation of current-level comparison across filter update (mediated cont.: splicing between different-filter patterns by vertical specification of match, although it includes lateral cross-comparison of skip-distant specifications).
However, filter update feedback is periodic, so it doesn’t form continuous cross-filter comparison patterns xPs.

adjustment of forward evaluation by optional feedback of projected input

More precisely, additive value or novel magnitude of an input is its deviation from higher-level average. Deviation = input - expectation: (higher-level summed input - summed difference /2) * rL (L / hL).
Inputs are compared to last input to form difference, and to past average to form deviation or novelty.

But last input is more predictive of the next one than a more distant average, thus the latter is compared on higher level than the former. So, input variable is compared sequentially and summed within resulting patterns. On the next level, the sum is compared vertically: to next-next-level average of the same variable.

Resulting vertical match defines novel value for higher-level sequential comparison:
novel value = past match - (vertical match * higher-level match rate) - average novel match:
nv = L+M - (m (I, (hI * rL)) * hM / hL) - hnM * rL; more precise than initial value: v = L+M - hM * rL;

Novelty evaluation is done if higher-level match > cost of feedback and operations, separately for I and D P s:
I, M ( D, M feedback, vertical SUB (I, nM ( D, ndM));
Impact on ambient sensor is separate from novelty and is predicted by representational-value patterns?

- next-input prediction: seq match + vert match * relative rate, but predictive selection is per level, not input.
- higher-order expectation is relative match per variable: pMd = D * rM, M/D, or D * rMd: Md/D,
- if rM | rMd are derived by intra-pattern comparison, when average M | Md > average per division?

one-input search extension within cross-compared patterns

Match decreases with distance, so initial comparison is between consecutive inputs. Resulting match is evaluated, forming ±vP s. Positive P s are then evaluated for expanded internal search: cross-comparison among 1-input-distant inputs within a pattern (on same level, higher-level search is between new patterns).

This cycle repeats to evaluate cross-comparison among 2-input-distant inputs, 3-input-distant inputs, etc., when summed current-distance match exceeds the average per evaluation.  
So, patterns of longer cross-comparison range are nested within selected positive patterns of shorter range. This is similar to 1st level ddP s being nested within dP s.

Same input is re-evaluated for comparison at increased distance because match will decay: projected match = last match * match rate (mr), * (higher-level mr / current-level mr) * (higher-level distance / next distance)?
Or = input * average match rate for that specific distance, including projected match within negative patterns.

It is re-evaluated also because projected match is adjusted by past match: mr *= past mr / past projected mr?
Also, multiple comparisons per input form overlapping and redundant patterns (similar to fuzzy clusters),
and must be evaluated vs. filter * number of prior comparisons, reducing value of projected match.

Instead of directly comparing incrementally distant input pairs, we can calculate their difference by adding intermediate differences. This would obviate multiple access to the same inputs during cross-comparison.
These differences are also subtracted (compared), forming higher derivatives and matches:

 ddd, x1dd, x2d  ( ddd: 3rd derivative,  x1dd: d of 2-input-distant d s,  x2d: d of 2-input-distant inputs)
                 /       \
 dd, x1d  dd, x1d  ( dd: 2nd derivative, x1d = d+d = difference between 1-input-distant inputs)
      /         \   /          \
   d       d          d      ( d: difference between consecutive inputs)
  /   \   /    \   /    \
i           i       i    i           ( i: initial inputs)

As always, match is a smaller input, cached or restored, selected by the sign of a difference.
Comparison of both types is between all same-type variable pairs from different inputs.
Total match includes match of all its derivation orders, which will overlap for proximate inputs.

Incremental cost of cross-comparison is the same for all derivation orders. If projected match is equal to projected miss, then additive value for different orders of the same inputs is also the same: reduction in projected magnitude of differences will be equal to reduction in projected match between distant inputs?

multi-input search extension, evaluation of selection per input: tentative

On the next level, average match from expansion is compared to that from shorter-distance comparison, and resulting difference is decay of average match with distance. Again, this decay drives re-evaluation per expansion: selection of inputs with projected decayed match above average per comparison cost.

Projected match is also adjusted by prior match (if local decay?) and redundancy (symmetrical if no decay?)
Slower decay will reduce value of selection per expansion because fewer positive inputs will turn negative:
Value of selection = Σ |comp cost of neg-value inputs| - selection cost (average saved cost or relative delay?)

This value is summed between higher-level inputs, into average value of selection per increment of distance. Increments with negative value of selection should be compared without re-evaluation, adding to minimal number of comparisons per selection, which is evaluated for feedback as a comparison-depth filter:

Σ (selection value per increment) -> average selection value;; for negative patterns of each depth, | >1 only?
depth adjustment value = average selection value; while (|average selection value| > selection cost){
depth adjustment ±±; depth adjustment value -= selection value per increment (depth-specific?); };
depth adjustment > minimal per feedback? >> lower-level depth filter;; additive depth = adjustment value?

- match filter is summed and evaluated per current comparison depth?
- selected positive relative matches don’t reduce the benefit of pruning-out negative ones.
- skip if negative selection value: selected positive matches < selection cost: average value or relative delay?

Each input forms a queue of matches and misses relative to templates within comparison depth filter. These derivatives, both discrete and summed, overlap for inputs within each other’s search span. But representations of discrete derivatives can be reused, redundancy is only necessary for parallel comparison.

Assuming that environment is not random, similarity between inputs declines with spatio-temporal distance. To maintain proximity, a n-input search is FIFO: input is compared to all templates up to maximal distance, then added to the queue as a new template, while the oldest template is outputted into pattern-wide queue.

value-proportional combination of patterns: tentative  

Summation of +dP and -dP is weighted by their value: L (summed d-sign match) + M (summed i match).
Such relative probability of +dP vs. - dP is indicated by corresponding ratios: rL = +L/-L, and rM = +M/-M.
(Ls and Ms are compared by division: comparison power should be higher for more predictive variables).

But weighting complementation incurs costs, which must be justified by value of ratio. So, division should be of variable length, continued while the ratio is above average. This is shown below for Ls, also applies to Ms:

dL = +L - -L, mL = min (+L, -L); nL =0; fL=0; efL=1; // nL: L multiple, fL: L fraction, efL: extended fraction.
while (dL > adL){ dL = |dL|; // all Ls are positive; dL is evaluated for long division by adL: average dL.
while (dL > 0){ dL -= mL; nL++;} dL -= mL/2; dL >0? fL+= efL; efL/=2;} // ratio: rL= nL + fL.

Ms’ long-division evaluation is weighted by rL: projected rM value = dM * nL (reduced-resolution rL) - adM.
Ms are then combined: cM = +M + -M * rL; // rL is relative probability of -M across iterated cL.
Ms are not projected (M+= D * rcL * rM D (MD/cD) /2): precision of higher-level rM D is below that of rM?

Prior ratios are combination rates: rL is probability of -M, and combined rL and rM (cr) is probability of -D.
If rM < arM, cr = rL, else: cr = (+L + +M) / (-L + -M) // cr = √(rL * rM) would lose L vs. M weighting.
cr predicts match of weighted cD between cdPs, where negative-dP variable is multiplied by above-average match ratio before combination: cD = +D + -D * cr. // after un-weighted comparison between Ds?

Averages: arL, arM, acr, are feedback of ratios that co-occur with above-average match of span-normalized variables, vs. input variables. Another feedback is averages that evaluate long division: adL, adM, adD.
Both are feedback of positive C pattern, which represents these variables, inputted & evaluated on 3rd level.
; or 4th level: value of dPs * ratio is compared to value of dPs, & the difference is multiplied by cL / hLe cL?

Comparison of opposite-sign Ds forms negative match = smaller |D|, and positive difference dD = +D+ |-D|.
dD magnitude predicts its match, not further combination. Single comparison is cheaper than its evaluation.
Comparison is by division if larger |D| co-occurs with hLe nD of above-average predictive value (division is sign-neutral & reductive). But average nD value is below the cost of evaluation, except if positive feedback?

So, default operations for L, M, D of complementary dPs are comparison by long division and combination.
D combination: +D -D*cr, vs. - cD * cr: +D vs. -D weighting is lost, meaningless if cD=0?
Combination by division is predictive if the ratio is matching on higher level (hLe) & acr is fed back as filter?            
Resulting variables: cL, rL, cM, rM, cr, cD, dD, form top level of cdP: complemented dP.

Level 3: prevaluation of projected filter patterns, forming updated-input patterns

(out of date)

3rd level inputs are ± V patterns, combined into complemented V patterns. Positive V patterns include derivatives of 1st level match, which project match within future inputs (D patterns only represent and project derivatives of magnitude). Such projected-inputs-match is pre-valuated, negative prevalue-span inputs are summed or skipped (reloaded), and positive prevalue-span inputs are evaluated or even directly compared.

Initial upward (Λ) prevaluation by E filter selects for evaluation of V patterns, within resulting ± E patterns. Resulting prevalue is also projected downward (V), to select future input spans for evaluation, vs. summation or skipping. The span is of projecting V pattern, same as of lower hierarchy. Prevaluation is then iterated over multiple projected-input spans, as long as last |prevalue| remains above average for the cost of prevaluation.

Additional interference of iterated negative projection is stronger than positive projection of lower levels, and should flush them out of pipeline. This flushing need not be final, spans of negative projected value may be stored in buffers, to delay the loss. Buffers are implemented in slower and cheaper media (tape vs. RAM) and accessed if associated patterns match on a higher level, thus project above-average match among their inputs.

Iterative back-projection is evaluated starting from 3rd level: to be projectable the input must represent derivatives of value, which are formed starting from 2nd level. Compare this to 2nd level evaluation:
Λ for input, V for V filter, iterated within V pattern. Similar sub-iteration in E pattern?

Evaluation value = projected-inputs-match - E filter: average input match that co-occurs with average  higher-level match per evaluation (thus accounting for evaluation costs + selected comparison costs). Compare this to V filter that selects for 2nd level comparison: average input match that co-occurs with average higher-level match per comparison (thus accounting for costs of default cross-comparison only).

E filter feedback starts from 4th level of search, because its inputs represent pre-valuated lower-level inputs.
4th level also pre-pre-valuates vs. prevaluation filter, forming pre-prevalue that determines prevaluation vs. summation of next input span. And so on: the order of evaluation increases with the level of search.
Higher levels are increasingly selective in their inputs, because they additionally select by higher orders derived on these levels: magnitude ) match and difference of magnitude ) match and difference of match, etc.

Feedback of prevaluation is ± pre-filter: binary evaluation-value sign that determines evaluating vs. skipping initial inputs within projected span, and flushing those already pipelined within lower levels.
Negative feedback may be iterated, forming a skip span.
Parallel lower hierarchies & skip spans may be assigned to different external sources or their internal buffers.

Filter update feedback is level-sequential, but pre-filter feedback is sent to all lower levels at once.
Pre-filter is defined per input, and then sequentially translated into pre-filters of higher derivation levels:
prior value += prior match -> value sign: next-level pre-filter. If there are multiple pre-filters of different evaluation orders from corresponding levels, they AND & define infra-patterns: sign ( input ( derivatives.

filter update evaluation and feedback

Negative evaluation-value blocks input evaluation (thus comparison) and filter updating on all lower levels. Not-evaluated input spans (gaps) are also outputted, which will increase coordinate range per contents of both higher-level inputs and lower-level feedback. Gaps represent negative projected-match value, which must be combined with positive value of subsequent span to evaluate comparison across the gap on a higher level. This is similar to evaluation of combined positive + negative relative match spans, explained above.

Blocking locations with expected inputs will result in preference for exploration & discovery of new patterns, vs. confirmation of the old ones. It is the opposite of upward selection for stronger patterns, but sign reversal in selection criteria is basic feature of any feedback, starting with average match & derivatives.

Positive evaluation-value input spans are evaluated by lower-level filter, & this filter is evaluated for update:
combined update = (output update + output filter update / (same-filter span (fL) / output span)) /2.
both updates: -= last feedback, equal-weighted because higher-level distance is compensated by range: fL?
update value = combined update - update filter: average update per average higher-level additive match.
also differential costs of feedback transfer across locations (vs. delay) + representation + filter conversion?

If update value is negative: fL += new inputs, subdivided by their positive or negative predictive value spans.
If update value is positive: lower-level filter += combined update, new fL (with new filter representation) is initialized on a current level, while current-level part of old fL is outputted and evaluated as next-level input.

In turn, the filter gets updates from higher-level outputs, included in higher-higher-level positive patterns by that level’s filter. Hence, each filter represents combined span-normalized feedback from all higher levels, of exponentially growing span and reduced update frequency.
Deeper hierarchy should block greater proportion of inputs. At the same time, increasing number of levels contribute to projected additive match, which may justify deeper search within selected spans.

Higher-level outputs are more distant from current input due to elevation delay, but their projection range is also greater. So, outputs of all levels have the same relative distance (distance/range) to a next input, and are equal-weighted in combined update. But if input span is skipped, relative distance of skip-initiating pattern to next input span will increase, and its predictive value will decrease. Hence, that pattern should be flushed or at least combined with a higher-level one:

combined V prevalue = higher-level V prevalue + ((current-level V prevalue - higher-level V prevalue) / ((current-level span / distance) / (higher-level span / distance)) /2. // the difference between current-level and higher-level prevalues is reduced by the ratio of their relative distances.

To speed up selection, filter updates can be sent to all lower levels in parallel. Multiple direct filter updates are span-normalized and compared at a target level, and the differences are summed in combined update. This combination is equal-weighted because all levels have the same span-per-distance to next input, where the distance is the delay of feedback during elevation. // this happens automatically in level-sequential feedback?

combined update = filter update + distance-normalized difference between output & filter updates:
((output update - filter update) / (output relative distance / higher-output relative distance)) /2.
This combination method is accurate for post-skipped input spans, as well as next input span.

- filter can also be replaced by output + higher-level filter /2, but value of such feedback is not known.
- possible fixed-rate sampling, to save on feedback evaluation if slow decay, ~ deep feedforward search?
- selection can be by patterns, derivation orders, sub-patterns within an order, or individual variables?
- match across distance also projects across distance: additive match = relative match * skipped distance?

cross-level shortcuts: higher-level sub-filters and symbols

After individual input comparison, if match of a current scale (length-of-a-length…) projects positive relative match of input lower-scale / higher-derivation level, then the later is also cross-compared between the inputs.
Lower scale levels of a pattern represent old lower levels of a search hierarchy (current or buffered inputs).

So, feedback of lower scale levels goes down to corresponding search levels, forming shortcuts to preserve detail for higher levels. Feedback is generally negative: expectations are redundant to inputs. But specifying feedback may be positive: lower-level details are novel to a pattern, & projected to match with it in the future.
Higher-span comparison power is increased if lower-span comparison match is below average:
variable subtraction ) span division ) super-span logarithm?

Shortcuts to individual higher-level inputs form a queue of sub-filters on a lower level, possibly represented by a queue-wide pre-filter. So, a level has one filter per parallel higher level, and sub-filter for each specified sub-pattern. Sub-filters of incrementally distant inputs are redundant to all previous ones.
Corresponding input value = match - sub-filter value * rate of match to sub-filter * redundancy?  

Shortcut to a whole level won’t speed-up search: higher-level search delay > lower-hierarchy search delay.
Resolution and parameter range may also increase through interaction of co-located counter-projections?

Symbols, for communication among systems that have common high-level concepts but no direct interface, are “co-author identification” shortcuts: their recognition and interpretation is performed on different levels.

Higher-level patterns have increasing number of derivation levels, that represent corresponding lower search levels, and project across multiple higher search levels, each evaluated separately?
Match across discontinuity may be due to additional dimensions or internal gaps within patterns.

Search depth may also be increased by cross-comparison between levels of scale within a pattern: match across multiple scale levels also projects over multiple higher- and lower- scale levels? Such comparison between variable types within a pattern would be of a higher order:

5. Comparison between variable types within a pattern (tentative)

To reiterate, elevation increases syntactic complexity of patterns: the number of different variable types within them. Syntax is identification of these types by their position (syntactic coordinate) within a pattern. This is analogous to recognizing parts of speech by their position within a sentence.
Syntax “synchronizes” same-type variables for comparison | aggregation between input patterns. Access is hierarchical, starting from sign->value levels within each variable of difference and relative match: sign is compared first, forming + and - segments, which are then evaluated for comparison of their values.

Syntactic expansion is pruned by selective comparison vs. aggregation of individual variable types within input patterns, over each coordinate type or resolution. As with templates, minimal aggregation span is resolution of individual inputs, & maximal span is determined by average magnitude (thus match) of new derivatives on a higher level. Hence, a basic comparison cycle generates queues of interlaced individual & aggregate derivatives at each template variable, and conditional higher derivatives on each of the former.

Sufficiently complex syntax or predictive variables will justify comparing across “syntactic“ coordinates within a pattern, analogous to comparison across external coordinates. In fact, that’s what higher-power comparisons do. For example, division is an iterative comparison between difference & match: within a pattern (external coordinate), but across derivation (syntactic coordinate).

Also cross-variable is comparison between orders of match in a pattern: magnitude, match, match-of-match... This starts from comparison between match & magnitude: match rate (mr) = match / magnitude. Match rate can then be used to project match from magnitude: match = magnitude * output mr * filter mr.
In this manner, mr of each match order adjusts intra-order-derived sequentially higher-order match:
match *= lower inter-order mr. Additive match is then projected from adjusted matches & their derivatives.

This inter-order projection continues up to the top order of match within a pattern, which is the ultimate selection criterion because that’s what’s left matching on the top level of search.
Inter-order vectors are ΛV symmetrical, but ΛV derivatives from lower order of match are also projected for higher-order match, at the same rate as the match itself?

Also possible is comparison across syntactic gaps: ΛY comparison -> difference, filter feedback VY hierarchy. For example, comparison between dimensions of a multi-D pattern will form possibly recurrent proportions.

Internal comparisons can further compress a pattern, but at the cost of adding a higher-order syntax, which means that they must be increasingly selective. This selection will increase “discontinuity” over syntactic coordinates: operations necessary to convert the variables before comparison. Eventually, such operators will become large enough to merit direct comparisons among them. This will produce algebraic equations, where the match (compression) is a reduction in the number of operations needed to produce a result.

The first such short-cut would be a version of Pythagorean theorem, discovered during search in 2D (part 6) to compute cosines. If we compare 2D-adjacent 1D Ls by division, over 1D distance and derivatives (an angle), partly matching ratio between the ratio of 1D Ls and a 2nd derivative of 1D distance will be a cosine.
Cosines are necessary to normalize all derivatives and lengths (Ls) to a value they have when orthogonal to 1D scan lines (more in part 6).

Such normalization for a POV angle is similar to dimensionality reduction in Machine Learning, but is much more efficient because it is secondary to selective dimensionality expansion. It’s not really “reduction”: dimensionality is prioritized rather than reduced. That is, the dimension of pattern’s main axis is maximized, and dimensions sequentially orthogonal to higher axes are correspondingly minimized. The process of discovering these axes is so basic that it might be hard-wired in animals.

6. Cartesian dimensions and sensory modalities

This is a recapitulation and expansion on incremental dimensionality introduced in part 2.
Term “dimension” here is reserved for a parameter that defines sequence and distance among inputs, initially Cartesian dimensions + Time. This is different from terminology of combinatorial search, where dimension is any parameter of an input, and their external order and distance don’t matter. My term for that is “variable“, external dimensions become types of a variable only after being encoded within input patterns.

For those with ANN background, I want to stress that a level of search in my approach is 1D queue of inputs, not a layer of nodes. The inputs to a node are combined regardless of difference and distance between them (the distance is the difference between laminar coordinates of source “neurons”).
These derivatives are essential because value of any prediction = precision of what * precision of where. Coordinates and co-derived differences are not represented in ANNs, so they can't be used to calculate Euclidean vectors. Without such vectors, prediction and selection of where must remain extremely crude.

Also, layers in ANN are orthogonal to the direction of input flow, so hierarchy is at least 2D. The direction of inputs to my queues is in the same dimension as the queue itself, which means that my core algorithm is 1D. A hierarchy of 1D queues is the most incremental way to expand search: we can add or extend only one coordinate at a time. This allows algorithm to select inputs that are predictive enough to justify the cost of representing additional coordinate and corresponding derivatives. Again, such incremental syntax expansion is my core principle, because it enables selective (thus scalable) search.

A common objection is that images are “naturally” 2D, and our space-time is 4D. Of course, these empirical facts are practically universal in our environment. But, a core cognitive algorithm must be able to discover and forget any empirical specifics on its own. Additional dimensions can be discovered as some general periodicity in the input flow: distances between matching inputs are compared, match between these distances indicates a period of lower dimension, and recurring periods form higher-dimension coordinate.

But as a practical shortcut to expensive dimension-discovery process, initial levels should be designed to specialize in sequentially higher spatial dimensions: 1D scan lines, 2D frames, 3D set of confocal “eyes“, 4D temporal sequence. These levels discover contiguous (positive match) patterns of increasing dimensionality:
1D line segments, 2D blobs, 3D objects, 4D processes. Higher 4D cycles form hierarchy of multi-dimensional orders of scale, integrated over time or distributed sensors. These higher cycles compare discontinuous patterns. Corresponding dimensions may not be aligned across cycles of different scale order.

Explicit coordinates and incremental dimensionality are unconventional. But the key for scalable search is input selection, which must be guided by cost-benefit analysis. Benefit is projected match of patterns, and cost is representational complexity per pattern. Any increase in complexity must be justified by corresponding increase in discovered and projected match of selected patterns. Initial inputs have no known match, thus must have minimal complexity: single-variable “what”, such as brightness of a grey-scale pixel, and single-variable “where”: pixel’s coordinate in one Cartesian dimension.

Single coordinate means that comparison between pixels must be contained within 1D (horizontal) scan line, otherwise their coordinates are not comparable and can’t be used to select locations for extended search. Selection for contiguous or proximate search across scan lines requires second (vertical) coordinate. That increases costs, thus must be selective according to projected match, discovered by past comparisons within 1D scan line. So, comparison across scan lines must be done on 2nd level of search. And so on.

Dimensions are added in the order of decreasing rate of change. This means spatial dimensions are scanned first: their rate of change can be sped-up by moving sensors. Comparison over purely temporal sequence is delayed until accumulated change / variation justifies search for additional patterns. Temporal sequence is the original dimension, but it is mapped on spatial dimensions until spatial continuum is exhausted. Dimensionality represented by patterns is increasing on higher levels, but each level is 1D queue of patterns.

Also independently discoverable are derived coordinates: any variable with cumulative match that correlates with combined cumulative match of all other variables in a pattern. Such correlation makes a variable useful for sequencing patterns before cross-comparison.
It is discovered by summing matches for same-type variables between input patterns, then cross-comparing summed matches between all variables of a pattern. Variable with the highest resulting match of match (mm) is a candidate coordinate. That mm is then compared to mm of current coordinate. If the difference is greater than cost of reordering future inputs, sequencing feedback is sent to lower levels or sensors.

Another type of empirically distinct variables is different sensory modalities: colors, sound and pitch, and so on, including artificial senses. Each modality is processed separately, up a level where match between patterns of different modalities but same scope exceeds match between unimodal patterns across increased distance. Subsequent search will form multi-modal patterns within common S-T frame of reference.

As with external dimensions, difference between modalities can be pre-defined or discovered. If the latter, inputs of different modalities are initially mixed, then segregated by feedback. Also as with dimensions, my core algorithm only assumes single-modal inputs, pre-defining multiple modalities would be an add-on.

7. Notes on clustering, ANNs, and probabilistic inference

In terms of conventional machine learning, my approach is a form of hierarchical fuzzy clustering. Cluster is simply a different term for pattern: a set of matching inputs. Each set is represented by a centroid: an input with below-threshold combined “distance” to other inputs of the set. The equivalent of centroid in my model is an input with above-average match (a complementary of a distance) to other inputs within its search span. Such inputs are selected to search next-level queue and to indirectly represent other cross-compared but not selected inputs, via their discrete or aggregate derivatives relative to the selected one.

Crucial differences here is that conventional clustering methods initialize centroids with arbitrary random weights, while I use matches (and so on) from past comparisons. And the weights are usually defined in terms of one variable, while I select higher-level inputs based on a combination of all variables per pattern, the number of which increases with the level of search.
Current methods in unsupervised learning were developed / accepted because they solved specific problems with reasonable resources. But, they aren’t comparable to human learning in scalability. I believe that requires an upfront investment in incremental complexity of representation: a syntactic overhead that makes such representation uncompetitive at short runs, but is necessary to predictively prune longer-range search.

The most basic example here is my use of explicit coordinates, and of input differences at their distances. I haven’t seen that in other low-level approaches, yet they are absolutely necessary to form Euclidean vectors. Explicit coordinates is why I start image processing with 1D scan lines, - another thing that no one else does. Images seem to be “naturally” 2D, but expanding search in 2Ds at once adds the extra cost of two sets of new coordinates & derivatives. On the other hand, adding 1D at a time allows to select inputs for each additional layer of syntax, reducing overall (number of variables * number of inputs) costs of search on the next level.

Artificial Neural Networks

The same coordinate-blind mindset pervades ANNs. Their learning is probabilistic: match is determined as an aggregate of multiple weighted inputs. Again, no derivatives (0D miss), or coordinates (1-4D miss) per individual input pair are recorded. Without them, there is no Euclidean vectors, thus pattern prediction must remain extremely crude. I use aggregation extensively, but this degradation of resolution is conditional on results of prior comparison between inputs. Neurons simply don’t have the capacity for primary comparison.

This creates a crucial difference in the way patterns are represented in the brain vs. my hierarchy of queues. Brain consists of neurons, each likely representing a variable of a given magnitude and modality. These variables are shared among multiple patterns, or co-activated networks (“cognits” in terms of J. Fuster).
This is conceptually perverse: relatively constant values of specific variables define a pattern, but there’s no reason that same values should be shared among different patterns. The brain is forced to share variables because it has fixed number of neurons, but a fluid and far greater number of their networks.

I think this is responsible for our crude taxonomy, such as using large, medium, small instead of specific numbers. So, it’s not surprising that our minds are largely irrational, even leaving aside all the sub-cortical nonsense. We don’t have to slavishly copy these constraints. Logically, a co-occurring set of variables should be localized within a pattern. This requires more local memory for redundant representations, but will reduce the need for interconnect and transfers to access global shared memory, which is far more expensive.

More broadly, neural network-centric mindset itself is detrimental, any function must be initially conceptualized as a sequential algorithm, parallelization is an optional superstructure.

Probabilistic inference: AIT, Bayesian logic, Markov models

A good introduction to Algorithmic information theory is Philosophical Treatise of Universal Induction by Rathmanner and Hutter. The criterion is same as mine: compression and prediction. But, while a progress vs. frequentist probability calculus, both AIT and Bayesian inference still assume a prior, which doesn’t belong in a consistently inductive approach. In my approach, priors or models are simply past inputs and their patterns. Subject-specific priors could speed-up learning, but unsupervised pattern discovery algorithm must be the core on which such shortcuts are added or removed from.

More importantly, as with most contemporary approaches, Bayesian learning is statistical and probabilistic. Probability is estimated from simple incidence of events, which I think is way too coarse. These events hardly ever match or miss precisely, so their similarity should be quantified. This would add a whole new dimension of micro-grayscale: partial match, as in my approach, vs. binary incidence in probabilistic inference. It should improve accuracy to the same extent that probabilistic inference improved on classical logic, by adding a macro-grayscale of partial probability vs. binary true | false values of the former.

Resolution of my inputs is always greater than that of their coordinates, while Bayesian inference and AIT typically start with the reverse: strings of 1bit inputs. These inputs, binary confirmations / disconfirmations, are extremely crude way to represent “events” or inputs. Besides, the events are assumed to be high-level concepts: the kind that occupy our conscious minds and are derived from senses by subconscious cognitive processes, which must be built into general algorithm. Such choice of initial inputs in BI and AIT demonstrates a lack of discipline in incrementing complexity.

To attempt a general intelligence, Solomonoff introduced “universal prior“: a class of all models. That class is a priori infinite, which means that he hits combinatorial explosion even *before* receiving actual inputs. It‘s a solution that only a mathematician may find interesting. Marginally practical implementation of AIT is Levin Search, which randomly generates models / algorithms of incremental complexity and selects those that happen to solve a problem or compress a bit string.

Again, I think starting with prior models is putting a cart before a horse: cognition must start with raw data, complex math only becomes cost-efficient on much higher levels of selection and generalization. And this distinction between input patterns and pattern discovery process is only valid within a level: algorithm is embedded in resulting patterns, hence is also compared on higher levels, forming “algorithmic patterns“.

8. Notes on working mindset and a prize for contributions

My terminology is as general as the subject itself. It’s a major confounder, - people crave context, but generalization is decontextualization. And cognitive algorithm is a meta-generalization: the only thing in common for everything we learn. This introduction is very compressed, because much of it is work in progress. But I think it also reflects and cultivates ruthlessly reductionist mindset required for such subject.

My math is very simple, because algorithmic complexity must be incremental. Advanced math can accelerate learning on higher levels of generalization, but it’s too expensive for initial levels. And general learning algorithm must be able to discover computational shortcuts (= math) on it’s own, just like we do. Complex math definitely isn’t innate in humans on any level: cavemen didn’t do calculus.

This work may seem too speculative, but any degree of generalization must be correspondingly lossy. Which is contrary to deduction-heavy culture of math and computer science. Hence, current Machine Learning is mostly experimental, with constrained tasks and supervised techniques. A handful of people aspire to work on AGI, but they either lack or neglect functional definition of intelligence, their theories are only vague inspiration.

I think working on this level demands greater delay of experimental verification than is acceptable in any established field. Except for philosophy, which has nothing else real to study. But established philosophers have always been dysfunctional fluffers, not surprisingly as their paying customers are college freshmen.

Our main challenge in formalizing GI is a specie-wide ADHD. We didn’t evolve for sustained focus on this level of generalization, that would cause extinction long before any tangible results. Which is no longer a risk, GI is the most important problem conceivable, and we have plenty of computing power for anything better than brute-force algorithms. But our psychology lags a light-year behind technology: we still hobble on mental crutches of irrelevant authority and peer support, flawed analogies and needless experimentation.

Prize for contributions

I offer prizes up to a total of $500K for debugging, optimizing, and extending this algorithm: github.
Contributions must fit into incremental-complexity hierarchy outlined here. Unless you find a flaw in my reasoning, which would be even more valuable. I can also pay monthly, but there must be a track record.
Winners will have an option to convert the awards into an interest in all commercial applications of a final algorithm, at the rate of $10K per 1% share. This would be informal and likely unprofitable, mine is not a commercial enterprise. Money can’t be primary motivation here, but it saves time.

Winners so far:

2010: Todor Arnaudov, $600 for suggestion to buffer old inputs after search. This occurred to me more than once before, but I rejected it as redundant to potential elevation of these inputs. Now that he made me think about it again, I realized that partial redundancy can preserve the detail at much lower cost than elevation.
The buffer is accessed if coordinates of contents get relatively close to projected inputs (that and justification is mine). It didn’t feel right because brain has no substrate for passive memory, but we do now.

2011: Todor, $400 consolation prize for understanding some ideas that were not clearly explained here.
2014: Dan He, $600 for pushing me to be more specific and to compare my algorithm with others.
2016: Todor, $500 for multiple suggestions on implementing the algorithm, as well as for the effort.
Kieran Greer, $375 for an attempt to implement my level 1 pseudo code in C#

2017: Alexander Loschilov, $2800 for help in converting my level 1 pseudo code into Python, consulting on PyCharm and SciPy, and for insistence on 2D clustering, February-April.
Todor: $2000 for help in optimizing level_1_2D, June-July.
Kapil Kashyap: $ 2000 for stimulation and effort, help with Python and level_1_2D, September-October