The On-Line Encyclopedia of Bongard Problems

Search: keyword:inductivedefinition

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Nested pairs of brackets vs. other arrangement of brackets (some open brackets are not closed or there are extra closing brackets).

(edit; present; nest [left/right]; search; history)

	COMMENTS	`Examples on the left are also known as "Dyck words".`
	REFERENCE	`https://en.wikipedia.org/wiki/Dyck_language`
	CROSSREFS	`Adjacent-numbered pages: BP951 BP952 BP953 BP954 BP955 * BP957 BP958 BP959 BP960 BP961`
	KEYWORD	`easy, nice, precise, allsorted, unwordable, notso, sequence, traditional, inductivedefinition, preciseworld, left-listable, right-listable`
	CONCEPT	`recursion (info \| search)`
	AUTHOR	`Aaron David Fairbanks`

BP1129

An oval is sorted left; shapes are sorted left when they can be built out of others sorted left by A) joining side by side (at a point) or B) joining one on top of the other (joining one's entire bottom edge to the other's entire top edge).

(edit; present; nest [left/right]; search; history)

	COMMENTS	`This was an unintended solution for BP1130.` `In category theory lingo, left examples are built by repeated horizontal composition and vertical composition. (Making horizontal lines as 0-ary vertical compositions is here forbidden.)`
	CROSSREFS	`Anything fitting right in BP1130 fits right here.` `Adjacent-numbered pages: BP1124 BP1125 BP1126 BP1127 BP1128 * BP1130 BP1131 BP1132 BP1133 BP1134`
	KEYWORD	`hard, less, convoluted, solved, inductivedefinition`
	CONCEPT	`or (info \| search)`
	WORLD	`[smaller \| same \| bigger]`
	AUTHOR	`Aaron David Fairbanks`

BP1200

The whole rectangle can be filled in by successively replacing pairs of adjacent rectangles with one vs. not so.

(edit; present; nest [left/right]; search; history)

	COMMENTS	`Another wording: "can be repeatedly broken along 'fault lines' to yield individual pieces vs not."`
	REFERENCE	`Robert Dawson, A forbidden suborder characterization of binarily composable diagrams in double categories, Theory and Applications of Categories, Vol. 1, No. 7, p. 146-145, 1995.`
	CROSSREFS	`All of the examples fitting left here would fit right in BP1199 except for (1) a single rectangle, (2) two rectangles stacked vertically, or (3) two rectangles side by side horizontally.` `All of the examples fitting right in in BP1097 (re-styled) would fit right here (besides a single solid block, but that isn't shown there).` `Adjacent-numbered pages: BP1195 BP1196 BP1197 BP1198 BP1199 * BP1201 BP1202 BP1203 BP1204 BP1205`
	KEYWORD	`hard, precise, challenge, proofsrequired, inductivedefinition, left-listable, right-listable`
	AUTHOR	`Aaron David Fairbanks`

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