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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

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

In an exact "less than __vs. greater than__" BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

As a rule of thumb, do not consider imperfections of hand drawn images when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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, it is still unknown where some examples would fit. Whether or not such a BP should be labelled "exact" 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 .)

NOTE: 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? Perhaps this, a fuzziness in the class of relevant examples but not the rule, should be its own independent keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule intends to sort. Different choices may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such choice. The label "exact" can only mean a Bongard Problem's rule seems precise to people who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? Perhaps this, a fuzziness in the class of relevant examples but not the rule, should be its own independent keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? Perhaps this, a fuzziness in the class of relevant examples but not the rule, should be its own independent keyword. - Aaron David Fairbanks, Nov 23 2021

Bongard Problems with precise definitions vs. Bongard Problems with vague definitions.

Bongard Problems with precise definitions vs. Bongard Problems with vagueness in the definition.

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own independent keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own independent keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

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

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

(Technical aside: to make matters murkier, some properties are known to be undecidable, and sometimes the decidability itself is unknown.)

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy; similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

(Technical aside: to make matters murkier, some properties are known to be undecidable, and sometimes the decidability itself unknown.)

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own keyword. - Aaron David Fairbanks, Nov 23 2021

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

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

In an exact BP, any relevant example is either clearly sorted left, clearly sorted right, or clearly not sorted. (No unsorted relevant cases at all is the keyword "allsorted" left-BP509.)

How can it be decided whether or not a rule is exact? How can it be decided whether or not a rule classifies all "examples that are relevant"? There needs to be another rule deciding which examples the original rule sorts. Different ways of formalizing that may have slightly different consequences; besides, Bongard Problems by design communicate ideas without fixing any such formalization. The label "exact" can only mean a BP's rule seems precise to those who see it. (This BP "exact vs. fuzzy" is fuzzy.)

In an exact spectrum-related BP (keyword "spectrum" left-BP507), the division between the sides is usually an apparent threshold. For example, there is an intuitive threshold between acute and obtuse angles (see BP292).

Imperfections in drawn images should generally not be considered when deciding whether a BP is exact or fuzzy. Just because one can draw a square badly does not mean "triangle vs. quadrilateral" (BP6) should be called fuzzy--similar vagueness arises in all hand-drawn BPs. (For Bongard Problems in which fine subtleties of drawings, including small imperfections, are meant to be considered, use the keyword "imperfectionscanmatter" left-BP913.)

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 used to verify a sorted example fits where it fits (some kind of mathematical proof); however, it is still unclear where some examples fit. Whether or not such a BP should be labelled "exact" might be debated.

(Technical aside: to make matters murkier, some properties are known to be undecidable, and sometimes the decidability itself unknown.)

NOTE: Sometimes the class of all examples in a Bongard Problem is imprecise, but, despite that fact, the underlying rule 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). Should a Bongard Problem like this be called "exact" or "fuzzy"? It is fuzziness in the class of relevant examples, but not the rule. Perhaps this should be its own keyword. - Aaron David Fairbanks, Nov 23 2021