Bongard Problems sorted left have the keyword "spectrum" on the OEBP.

In a "spectrum" Bongard Problem, there is an evident way to assign each object a value (e.g. "size" or "number of holes"). Then, to determine whether an object fits left or right in the Bongard Problem, its value is compared with a fixed threshold value.

Spectra can be continuous or discrete.

A "spectrum" Bongard Problem is usually arbitrary, since there could be made many different versions of it with different choices of threshold value. However, sometimes a certain choice of threshold is particularly natural. For example, the threshold of 90 degrees in "acute vs. obtuse angles" does not come across as arbitrary. And in BP2, the spectrum of values ("size") is vague, so much that the fuzzy threshold, of about half the size of the bounding box, does not seem arbitrary.

A spectrum Bongard Problem may or may not have the following properties:

1) The values assigned to objects are precise.

2) The threshold value between the two sides is precise.

3) The threshold value is itself sorted on one of the two sides.

Each of the latter two typically only makes sense when the condition before it is true.

If a spectrum Bongard Problem obeys 1) and 2), then it will usually be precise.

For example:

"Angles less than 90° vs. angles greater than 90°."

If a spectrum Bongard Problem obeys 1), 2), and 3), then it will usually be allsorted.

For example:

"Angles less than or equal to 90° vs. angles greater than 90°."

In a discrete spectrum Bongard Problem, even if it is precise, there isn't one unambiguous threshold value. Consider "2 or fewer holes vs. 3 or more holes". (Is the threshold 2? 3? 2.5?)

In an especially extreme kind of spectrum Bongard Problem, one side represents just a single value, just the threshold value. For example, "right angles vs. obtuse angles." In certain cases like this the threshold is an extreme value at the very boundary of the spectrum of possible values. For example, consider "no holes vs. one or more holes." Cases like this might not even be understood as two sides of a spectrum, but rather the absence versus presence of a property. (See the keyword notso.)

Even more extreme, in some Bongard Problems, each of the sides is a single value on a spectrum. For example, BP6 is "3 sides vs. 4 sides". We have not been labeling Bongard Problems like this with the keyword "spectrum".

After all, any Bongard Problem can be re-interpreted as a spectrum Bongard Problem, where the spectrum ranges from the extreme of fitting left to the extreme of fitting right.

https://en.wikipedia.org/wiki/Total_order

See BP874 for the version sorting pictures of Bongard Problems (miniproblems) instead of links to pages on the OEBP.

Adjacent-numbered pages:
BP502 BP503 BP504 BP505 BP506  *  BP508 BP509 BP510 BP511 BP512

notso, meta (see left/right), links, keyword, sideless

bp [smaller | same | bigger]
zoom in left (spectrum_bp)

Aaron David Fairbanks

Bongard Problems sorted left have the keyword "precise" on the OEBP.

Bongard Problems sorted right have the keyword "fuzzy" on the OEBP.

In an precise Bongard Problem, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted.

(All relevant examples clearly sorted either left or right is the keyword allsorted.)

How can it be decided whether or not a rule is precise? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule to determine which examples the original rule intends to sort. Bongard Problems by design communicate ideas without fixing that context ahead of time. The label "precise" can only mean a Bongard Problem's rule seems precise to people who see it. (This "precise vs. fuzzy" Bongard Problem is fuzzy.)

In an precise "less than ___ vs. greater than ___" Bongard Problem (keyword spectrum), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see e.g. BP292).

As a rule of thumb, do not consider imperfections of hand drawn images (keyword ignoreimperfections) when deciding whether a Bongard Problem is precise or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be labelled fuzzy; similar vagueness arises in all hand-drawn Bongard Problems. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword perfect.)

Sometimes the way a Bongard Problem would sort certain examples is an unsolved problem in mathematics. (See e.g. BP820.) There is a precise criterion that has been used to verify each sorted example fits where it fits (some kind of mathematical proof); however, where some examples fit is still unknown. Whether or not such a Bongard Problem should be labelled "precise" might be debated.

(Technical note: some properties are known to be undecidable, and sometimes the decidability itself is unknown. See https://en.wikipedia.org/wiki/Decision_problem .)

(See the keyword proofsrequired.)

One way to resolve this ambiguity is to define "precise" as meaning that once people decide where an example belongs for a reason, they will all agree about it.

Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that, the rule sorting those examples is precise. Say, for some potential new example, it is unclear whether it should be included in the Bongard Problem at all, but, if it were included, it would be clear where it should be sorted (or that it should be left unsorted). A Bongard Problem like this can still be tagged "precise".

(If all examples are clearly sorted except for some example for which it is unclear whether it belongs to the class of relevant examples, the situation becomes ambiguous.)

On the other hand, sometimes the class of all examples is very clear, with an obvious boundary. (Keyword preciseworld.)

There is a subtle distinction to draw between Bongard Problems that are precise to the people making them and Bongard Problems that are precise to the people solving them. A Bongard Problem (particularly a non-allsorted one) might be labeled "precise" on the OEBP because the description and the listed ambiguous examples explicitly forbid sorting certain border cases; however, someone looking at the Bongard Problem without access to the OEBP page containing the definition would not be aware of this. It may or may not be obvious that certain examples were intentionally left out of the Bongard Problem. A larger collection of examples may make it more clear that a particularly blatant potential border case was left out intentionally.

See BP876 for the version with pictures of Bongard Problems instead of links to pages on the OEBP.

See both and neither for specific ways an example can be classified as unsorted in an "precise" Bongard Problem.

Adjacent-numbered pages:
BP503 BP504 BP505 BP506 BP507  *  BP509 BP510 BP511 BP512 BP513

fuzzy, meta (see left/right), links, keyword, right-self, sideless

bp [smaller | same | bigger]

Aaron David Fairbanks

This is the keyword "dual" on the OEBP.

Given an example there is some way to "flip sides" by altering it. The left-to-right and right-to-left transformations should be inverses.

It is not required that there only be one such transformation. For example, for many handed Bongard Problem, flipping an example over any axis will reliably switch its sorting.

It is not required that every left example must have its corresponding right example uploaded on the OEBP nor vice versa. See the keyword contributepairs for the BPs the OEBP advises users upload left and right examples for in pairs.

Adjacent-numbered pages:
BP529 BP530 BP531 BP532 BP533  *  BP535 BP536 BP537 BP538 BP539

meta (see left/right), links, keyword, sideless

bp [smaller | same | bigger]

Aaron David Fairbanks

Left examples have the keyword "leftright" on the OEBP.

See updown.

All "leftright" Bongard Problems are handed.

Adjacent-numbered pages:
BP530 BP531 BP532 BP533 BP534  *  BP536 BP537 BP538 BP539 BP540

meta (see left/right), links, keyword, sideless, invariance

handed_visualbp [smaller | same | bigger]

Aaron David Fairbanks

Left examples have the keyword "experimental" on the OEBP.

Right examples have the keyword "traditional" on the OEBP.

Experimental BPs push the boundaries of what makes Bongard Problems Bongard Problems.

Traditional BPs show some simple property of black and white pictures. The OEBP is a place with many wild and absurd Bongard Problems, so it is useful to have an easy way to just find the regular old Bongard Problems.

Adjacent-numbered pages:
BP545 BP546 BP547 BP548 BP549  *  BP551 BP552 BP553 BP554 BP555

subjective, meta (see left/right), links, keyword, left-it

bp [smaller | same | bigger]

Aaron David Fairbanks

Left examples have the keyword "handed" on the OEBP.

If mirroring any example along the any axis can change its sorting the BP is "handed."

Note that BPs about comparing orientation between multiple things in one example fit on the right side.

See BP871 for the version with pictures of Bongard Problems (miniproblems) instead of links to pages on the OEBP.

The keyword leftright is specifically about flipping over the vertical axis, while the keyword updown is specifically about flipping over the horizontal axis.

Bongard Problems tagged rotate are usually "handed", since any rotation can be created by two reflections. Not necessarily, however, since the reflected step in between might not be sorted on either side by the Bongard Problem.

Adjacent-numbered pages:
BP547 BP548 BP549 BP550 BP551  *  BP553 BP554 BP555 BP556 BP557

meta (see left/right), links, keyword, invariance, wellfounded

visualbp [smaller | same | bigger]
zoom in left (handed_visualbp)

Aaron David Fairbanks

Left-sorted examples have the keyword "blackwhite" on the OEBP.

Right-sorted examples have the keyword "blackwhiteinvariant" on the OEBP.

All examples are visual Bongard Problems that allow black to touch the bounding box (keyword bordercontent).

Adjacent-numbered pages:
BP551 BP552 BP553 BP554 BP555  *  BP557 BP558 BP559 BP560 BP561

meta (see left/right), links, keyword, invariance, wellfounded

visualbp [smaller | same | bigger]

Leo Crabbe

Left examples have the keyword "boundingbox" on the OEBP.

Slightly different: sliding the content in a box around without letting it cross the bounding box and without changing the size of the bounding box. (See keyword absoluteposition.)

Expanding the boxes of BP2 ("big vs. small") makes the contents smaller in comparison to the box, but not smaller in an absolute sense. Hence the situation is ambiguous.

If a Bongard Problem has the keyword absoluteposition, then it likely has the keyword boundingbox.

If a Bongard Problem has the keyword boundingbox and does not have the keyword bordercontent, then it likely has the keyword absoluteposition.

Adjacent-numbered pages:
BP969 BP970 BP971 BP972 BP973  *  BP975 BP976 BP977 BP978 BP979

meta (see left/right), links, keyword, invariance

Aaron David Fairbanks

Left-sorted Bongard Problems have the keyword "updown" on the OEBP.

See leftright.

All "updown" Problems are handed.

Adjacent-numbered pages:
BP1004 BP1005 BP1006 BP1007 BP1008  *  BP1010 BP1011 BP1012 BP1013 BP1014

meta (see left/right), links, keyword, invariance

handed_visualbp [smaller | same | bigger]

Aaron David Fairbanks

This classification is specifically concerned with changes to examples that leave them sortable, as there are almost always ways of adding details to a BP's examples that make them unsortable.

Right-sorted BPs in this Bongard Problem are often Bongard Problems where there is always a way of adding to left-sorted examples to make them right-sorted, but not the other way around, or vice versa.

Another version of this Bongard Problem could be made about adding white (erasure of detail) instead of black (addition of detail).

Another version could be made about adding either white or black, but not both.

Where appropriate, you can assume all images will have some room in a lip of white background around the border (ignoring https://en.wikipedia.org/wiki/Sorites_paradox ).

You can't expand the boundary of an image as you add detail to it. If image boundaries could be expanded, then any shape could be shrunken to a point in relation to the surrounding whiteness, which could then be filled in to make any other shape.

How should this treat cases in which just a few examples can't be added to at all (like an all-black box)? E.g. BP966. Should this be sorted right (should the one special case of a black box spoil it) or should it be sorted left (should examples that can't at all be further added be discounted)? Maybe we should only sort BPs in which all examples can be further added to. (See BP1143^left.) - Aaron David Fairbanks, Nov 12 2021

Is "addition of detail" context-dependent, or does it just mean any addition of blackness to the image? Say you have a points-and-lines Bongard Problem like BP1100, and you're trying to decide whether to sort it left or right here. You would just want to think about adding more points and lines to the picture. You don't want to get bogged down in thinking about whether black could be added to the image in a weird way so that a point gets turned into a line, or something. - Aaron David Fairbanks, Nov 13 2021

See BP1139 for Bongard Problems in which no example can be added to, period.

Adjacent-numbered pages:
BP1134 BP1135 BP1136 BP1137 BP1138  *  BP1140 BP1141 BP1142 BP1143 BP1144

meta (see left/right), links, sideless

Leo Crabbe