From 6e129a43122508f39228d01821fdfcec392af90a Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sat, 17 May 2025 23:05:57 +0200 Subject: [PATCH 01/17] Added seq as whole object page --- .../ideas/sequence_as_a_whole_object.md | 24 +++++++++++++++++++ .../ideas/sequence_as_an_system_object.md | 3 --- mkdocs.yml | 2 +- 3 files changed, 25 insertions(+), 4 deletions(-) create mode 100644 docs/fundamentals/ideas/sequence_as_a_whole_object.md delete mode 100644 docs/fundamentals/ideas/sequence_as_an_system_object.md diff --git a/docs/fundamentals/ideas/sequence_as_a_whole_object.md b/docs/fundamentals/ideas/sequence_as_a_whole_object.md new file mode 100644 index 00000000..73e1227e --- /dev/null +++ b/docs/fundamentals/ideas/sequence_as_a_whole_object.md @@ -0,0 +1,24 @@ +# Sequence as a whole object + +Symbol sequences are a common model in many theoretical and applied science areas. +This prevalence is since almost any object of study can be represented as a sequence of elements or events. + +However, if we set aside our knowledge of the essence of the elements in a specific case and +consider the sequence as a separate object of study, then only probabilistic (statistical) methods or +various methods for comparing sequences (for example, the Levenshtein distance) are used. + +None of these methods describes the sequence as a holistic object. +The Levenshtein distance requires a different sequence to compare with the original one, which makes +the measures of this approach "relative". The probabilistic approach decomposes the sequence into +elements and calculates their probabilities. Thus, the sequence is replaced, as an object of study, +by a probability distribution. In turn, a specific probability distribution corresponds to an infinite +number of sequences with a ratio of elements "close" to the one in the original sequence. +Moments, conditional probabilities, Shannon entropy, and Markov chains allow us to more accurately +model the object under study, but they still essentially rely on the idea of ​​decomposing a sequence +into independent elements, ignoring the sequence as a holistic object. Practically all existing approaches +to the study and description of symbolic sequences originate from the set-theoretic approach. + +Formal order analysis is based on the belief that a symbolic sequence can be considered as a holistic object +with emergent properties, which corresponds to systems thinking. This method studies, in addition to the set +of elements, the internal structure (composition, pattern) of the sequence, which determines its uniqueness +among others, including those consisting of the same set of elements. diff --git a/docs/fundamentals/ideas/sequence_as_an_system_object.md b/docs/fundamentals/ideas/sequence_as_an_system_object.md deleted file mode 100644 index c46c29ec..00000000 --- a/docs/fundamentals/ideas/sequence_as_an_system_object.md +++ /dev/null @@ -1,3 +0,0 @@ -# Sequence as a System Object - -Coming soon diff --git a/mkdocs.yml b/mkdocs.yml index 8975d29b..2fbb0fdd 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -6,7 +6,7 @@ nav: - fundamentals/index.md - "Ideas": - fundamentals/ideas/index.md - - "Sequence as a whole object": fundamentals/ideas/sequence_as_an_system_object.md + - "Sequence as a whole object": fundamentals/ideas/sequence_as_a_whole_object.md - "Order as a sequence property": fundamentals/ideas/order_as_a_sequence_property.md - "Congeneric decomposition": fundamentals/ideas/congeneric_decomposition.md - "Interval as a basic information unit": fundamentals/ideas/interval_as_a_basic_information_unit.md From 3d8147b8b5ed561071f5057ddb1328558d4eb5ee Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 18 May 2025 20:02:07 +0200 Subject: [PATCH 02/17] Added order as a property pag --- .../fundamentals/ideas/order_as_a_property.md | 63 ++++++++++++++++++- .../ideas/sequence_as_a_whole_object.md | 2 +- mkdocs.yml | 2 +- 3 files changed, 64 insertions(+), 3 deletions(-) diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md index 19deeea3..452fa957 100644 --- a/docs/fundamentals/ideas/order_as_a_property.md +++ b/docs/fundamentals/ideas/order_as_a_property.md @@ -1,3 +1,64 @@ +--- +hide: + - toc +--- # Order as a Property -Coming soon +Formal order analysis defines a special property of symbolic sequences - an Order. +The order is a sequence of natural numbers obtained from the original symbolic sequence by replacing each +of its elements with a natural number corresponding to the index of this element in the alphabet +sorted by the appearance of the elements in the original sequence. + +The concept of an Order can be conveniently demonstrated using an example: + + +Let's assume there is a symbolic sequence "INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE" + +Highlight the first appearance of new elements "**I****N****T****E****L**LI**G**EN**C**E**_**I**S** T**H**E **A****B**ILIT**Y** T**O** A**D**A**P**T TO CHANGE" + +Enumirate first apperance elements + + + +| I | N | T | E | L | G | C | | S | H | A | B | Y | O | D | P | +|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----| +| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | + +Determine the order of the sequence by replacing each element of the sequence with its corresponding index + + +| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | +|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| +| 1 | 2 | 3 | 4 | 5 | 5 | 1 | 6 | 4 | 2 | 7 | 4 | 8 | 1 | 9 | 8 | 3 | 10 | 4 | 8 | 11 | 12 | 1 | 5 | 1 | 3 | 13 | 8 | 3 | 14 | 8 | 11 | 15 | 11 | 16 | 3 | 8 | 3 | 14 | 8 | 7 | 10 | 11 | 2 | 6 | 4 | + + +Despite the triviality of the concept Order, it allows to separate +the elements and composition of a sequence, to determine the compositional equivalence of different sequences. + +Example of sequences with equals orders: + +```pyodide exec="on" install="foapy,numpy" +import foapy +import numpy as np + +seqA = list("INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE") +seqB = list("1N73LL1G3NC321527H324B1L17Y27024D4P72702CH4NG3") +orderA = foapy.order(seqA) +orderB = foapy.order(seqB) +print("SeqA and SeqB orders are equals -", np.all(orderA == orderB)) +print("Order =", orderA) +``` diff --git a/docs/fundamentals/ideas/sequence_as_a_whole_object.md b/docs/fundamentals/ideas/sequence_as_a_whole_object.md index 73e1227e..d6c2ae12 100644 --- a/docs/fundamentals/ideas/sequence_as_a_whole_object.md +++ b/docs/fundamentals/ideas/sequence_as_a_whole_object.md @@ -20,5 +20,5 @@ to the study and description of symbolic sequences originate from the set-theore Formal order analysis is based on the belief that a symbolic sequence can be considered as a holistic object with emergent properties, which corresponds to systems thinking. This method studies, in addition to the set -of elements, the internal structure (composition, pattern) of the sequence, which determines its uniqueness +of elements, the internal structure (pattern) of the sequence, which determines its uniqueness among others, including those consisting of the same set of elements. diff --git a/mkdocs.yml b/mkdocs.yml index 2fbb0fdd..415f89e1 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -7,7 +7,7 @@ nav: - "Ideas": - fundamentals/ideas/index.md - "Sequence as a whole object": fundamentals/ideas/sequence_as_a_whole_object.md - - "Order as a sequence property": fundamentals/ideas/order_as_a_sequence_property.md + - "Order as a property": fundamentals/ideas/order_as_a_property.md - "Congeneric decomposition": fundamentals/ideas/congeneric_decomposition.md - "Interval as a basic information unit": fundamentals/ideas/interval_as_a_basic_information_unit.md - "Geomteric mean as alternative to probability": fundamentals/ideas/geometric_mean_based_characteristics.md From edabe8281b606a5baa915bab92729ae530e177f0 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 18 May 2025 22:31:01 +0200 Subject: [PATCH 03/17] Added order as a property pag --- docs/fundamentals/ideas/order_as_a_property.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md index 452fa957..9a780119 100644 --- a/docs/fundamentals/ideas/order_as_a_property.md +++ b/docs/fundamentals/ideas/order_as_a_property.md @@ -27,8 +27,7 @@ Enumirate first apperance elements } .md-typeset td, .md-typeset th { - padding-left: 0.4em !important; - padding-right: 0.4em !important; + padding: 0.1em 0.4 !important; text-align: center !important; } From aadf7e35979aa8bc3265b5206a7aab2f7c5006e1 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 18 May 2025 22:34:27 +0200 Subject: [PATCH 04/17] Added order as a property pag --- docs/fundamentals/ideas/order_as_a_property.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md index 9a780119..9fff3d07 100644 --- a/docs/fundamentals/ideas/order_as_a_property.md +++ b/docs/fundamentals/ideas/order_as_a_property.md @@ -14,7 +14,12 @@ The concept of an Order can be conveniently demonstrated using an example: Let's assume there is a symbolic sequence "INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE" -Highlight the first appearance of new elements "**I****N****T****E****L**LI**G**EN**C**E**_**I**S** T**H**E **A****B**ILIT**Y** T**O** A**D**A**P**T TO CHANGE" +Highlight the first appearance of new elements + +| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | +|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| +| 1 | 2 | 3 | 4 | 5 | | | 6 | | | 7 | | 8 | | 9 | | | 10 | | | 11 | 12 | | | | | 13 | | | 14 | | | 15 | | 16 | | | | | | | | | | | | + Enumirate first apperance elements +| 1 | 2 | 3 | 4 | 5 | | | 6 | | | 7 | | 8 | | 9 | | | 10 | | | 11 | 12 | | | | | 13 | | | 14 | | | 15 | | 16 | | | | | | | | | | | | +The alphabet for the sequence would be sequence of unique elements: | I | N | T | E | L | G | C | | S | H | A | B | Y | O | D | P | |---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | -Determine the order of the sequence by replacing each element of the sequence with its corresponding index + +Determine the order of the sequence by replacing each element of the sequence with its corresponding alphabet index | I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | 1 | 2 | 3 | 4 | 5 | 5 | 1 | 6 | 4 | 2 | 7 | 4 | 8 | 1 | 9 | 8 | 3 | 10 | 4 | 8 | 11 | 12 | 1 | 5 | 1 | 3 | 13 | 8 | 3 | 14 | 8 | 11 | 15 | 11 | 16 | 3 | 8 | 3 | 14 | 8 | 7 | 10 | 11 | 2 | 6 | 4 | +The order of symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE` is -Despite the triviality of the concept Order, it allows to separate -the elements and composition of a sequence, to determine the compositional equivalence of different sequences. +| 1 | 2 | 3 | 4 | 5 | 5 | 1 | 6 | 4 | 2 | 7 | 4 | 8 | 1 | 9 | 8 | 3 | 10 | 4 | 8 | 11 | 12 | 1 | 5 | 1 | 3 | 13 | 8 | 3 | 14 | 8 | 11 | 15 | 11 | 16 | 3 | 8 | 3 | 14 | 8 | 7 | 10 | 11 | 2 | 6 | 4 | +|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| + + + +Despite the triviality of the concept Order, it allows us to separate the elements and composition of a sequence and to define the compositional equivalence of different sequences. Example of sequences with equals orders: @@ -66,3 +56,21 @@ orderB = foapy.order(seqB) print("SeqA and SeqB orders are equals -", np.all(orderA == orderB)) print("Order =", orderA) ``` + + From ea5d3097b89b72c3e56c1117a79f3460e5010cb2 Mon Sep 17 00:00:00 2001 From: Nikolay Pozdnichenko Date: Tue, 20 May 2025 00:53:28 +0600 Subject: [PATCH 06/17] Update sequence_as_a_whole_object.md Fixed some phrases and sentences. --- .../ideas/sequence_as_a_whole_object.md | 22 ++++++++++--------- 1 file changed, 12 insertions(+), 10 deletions(-) diff --git a/docs/fundamentals/ideas/sequence_as_a_whole_object.md b/docs/fundamentals/ideas/sequence_as_a_whole_object.md index d6c2ae12..69872ac6 100644 --- a/docs/fundamentals/ideas/sequence_as_a_whole_object.md +++ b/docs/fundamentals/ideas/sequence_as_a_whole_object.md @@ -1,24 +1,26 @@ # Sequence as a whole object -Symbol sequences are a common model in many theoretical and applied science areas. +Texts / ordered data arrays / time series of various nature are a common model in theoretical and applied science. This prevalence is since almost any object of study can be represented as a sequence of elements or events. However, if we set aside our knowledge of the essence of the elements in a specific case and -consider the sequence as a separate object of study, then only probabilistic (statistical) methods or -various methods for comparing sequences (for example, the Levenshtein distance) are used. +consider the sequence itself as a separate object of study, then it turns out that methods used +to study them are almost exclusively statistical. +With the only exception being methods for comparison / alignment of two or more sequences +(for example, the Levenshtein distance). -None of these methods describes the sequence as a holistic object. -The Levenshtein distance requires a different sequence to compare with the original one, which makes +None of these methods describe a sequence as a holistic object. +The Levenshtein distance requires another sequence to compare with the original one, which makes the measures of this approach "relative". The probabilistic approach decomposes the sequence into -elements and calculates their probabilities. Thus, the sequence is replaced, as an object of study, +elements and calculates their probabilities (frequencies). Thus, the sequence is replaced, as an object of study, by a probability distribution. In turn, a specific probability distribution corresponds to an infinite number of sequences with a ratio of elements "close" to the one in the original sequence. Moments, conditional probabilities, Shannon entropy, and Markov chains allow us to more accurately -model the object under study, but they still essentially rely on the idea of ​​decomposing a sequence +model the object under study, but they still essentially rely on the idea of decomposing a sequence into independent elements, ignoring the sequence as a holistic object. Practically all existing approaches to the study and description of symbolic sequences originate from the set-theoretic approach. Formal order analysis is based on the belief that a symbolic sequence can be considered as a holistic object -with emergent properties, which corresponds to systems thinking. This method studies, in addition to the set -of elements, the internal structure (pattern) of the sequence, which determines its uniqueness -among others, including those consisting of the same set of elements. +with emergent properties, which corresponds to systems thinking. In addition to the distribution +of elements this method studies arrangement of its components - the internal structure (pattern) of the sequence, +which determines its uniqueness among others, including those consisting of the same set of elements. From 22a309717cafe2be4c634f016ce70f62e0c6c32e Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sat, 24 May 2025 10:01:13 +0200 Subject: [PATCH 07/17] Fix text --- docs/fundamentals/ideas/sequence_as_a_whole_object.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/fundamentals/ideas/sequence_as_a_whole_object.md b/docs/fundamentals/ideas/sequence_as_a_whole_object.md index 69872ac6..3f4b72dd 100644 --- a/docs/fundamentals/ideas/sequence_as_a_whole_object.md +++ b/docs/fundamentals/ideas/sequence_as_a_whole_object.md @@ -1,12 +1,12 @@ # Sequence as a whole object -Texts / ordered data arrays / time series of various nature are a common model in theoretical and applied science. +Symbol sequences are a common model in theoretical and applied science. This prevalence is since almost any object of study can be represented as a sequence of elements or events. However, if we set aside our knowledge of the essence of the elements in a specific case and -consider the sequence itself as a separate object of study, then it turns out that methods used -to study them are almost exclusively statistical. -With the only exception being methods for comparison / alignment of two or more sequences +consider the sequence itself as a separate object of study, then it turns out that methods used +to study them are almost exclusively statistical. +With the only exception being methods for comparison / alignment of two or more sequences (for example, the Levenshtein distance). None of these methods describe a sequence as a holistic object. @@ -22,5 +22,5 @@ to the study and description of symbolic sequences originate from the set-theore Formal order analysis is based on the belief that a symbolic sequence can be considered as a holistic object with emergent properties, which corresponds to systems thinking. In addition to the distribution -of elements this method studies arrangement of its components - the internal structure (pattern) of the sequence, +of elements this method studies arrangement of its components - the internal structure (pattern) of the sequence, which determines its uniqueness among others, including those consisting of the same set of elements. From ab112126a4e86ccc0451c835640625ca50f4d727 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 1 Jun 2025 13:40:55 +0200 Subject: [PATCH 08/17] Added Congeneric decomposition --- .../ideas/congeneric_decomposition.md | 63 ++++++++++++++++++- 1 file changed, 62 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/congeneric_decomposition.md b/docs/fundamentals/ideas/congeneric_decomposition.md index 878292d3..bedc9037 100644 --- a/docs/fundamentals/ideas/congeneric_decomposition.md +++ b/docs/fundamentals/ideas/congeneric_decomposition.md @@ -1,3 +1,64 @@ +--- +hide: + - toc +--- # Congeneric Decomposition -Coming soon +Cogeneric decomposition is a method for decomposing symbolic sequences from a systems thinking perspective, +emphasizing the importance of order. It decomposes a sequence into a tuple of cogeneric sequences, +each of which consists of equivalent elements at certain positions, while all other positions are empty. +This reversible process preserves the order of the sequence and allows the original sequence to be fully reconstructed. + +The concept of Cogeneric decomposition can be demonstrated using an example: + +Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE` congeneric decomposition +could be presented by the following table, where each row is a congeneric sequence and `-` is an empty position in a congeneric sequence. + +| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | +|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| +| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +| I | - | - | - | - | - | I | - | - | - | - | - | - | I | - | - | - | - | - | - | - | - | I | - | I | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | N | - | - | - | - | - | - | - | N | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | N | - | - | +| - | - | T | - | - | - | - | - | - | - | - | - | - | - | - | - | T | - | - | - | - | - | - | - | - | T | - | - | T | - | - | - | - | - | - | T | - | T | - | - | - | - | - | - | - | - | +| - | - | - | E | - | - | - | - | E | - | - | E | - | - | - | - | - | - | E | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | E | +| - | - | - | - | L | L | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | L | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | G | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | G | - | +| - | - | - | - | - | - | - | - | - | - | C | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | C | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - |    | - | - |    | - | - | - |    | - | - | - | - | - | - | - |    | - | - |    | - | - | - | - | - |    | - | - |    | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | S | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | H | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | H | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | A | - | - | - | - | - | - | - | - | - | - | A | - | A | - | - | - | - | - | - | - | - | A | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | B | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | Y | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | O | - | - | - | - | - | - | - | - | O | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | D | - | - | - | - | - | - | - | - | - | - | - | - | - | +| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | P | - | - | - | - | - | - | - | - | - | - | - | + + +Congeneric sequence for `I` +``` +I - - - - - I - - - - - - I - - - - - - - - I - I - - - - - - - - - - - - - - - - - - - - - +``` +could be a part of multiple symbol sequences that have the same order of `I` element. + +While keeping the main idea, the congeneric decomposition could be applied, with a flavor, to any type of special case symbolic sequences, such as Order. + + + + From bbc30060d333a2edb0b9204e2794dda90bfe09a1 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 8 Jun 2025 00:24:19 +0200 Subject: [PATCH 09/17] Interval as information unit --- .pre-commit-config.yaml | 1 + .../interval_as_a_basic_information_unit.md | 43 ++++++++++++++++++- mkdocs.yml | 6 ++- 3 files changed, 48 insertions(+), 2 deletions(-) diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml index 0250b21d..e8abf785 100644 --- a/.pre-commit-config.yaml +++ b/.pre-commit-config.yaml @@ -11,6 +11,7 @@ repos: - id: check-merge-conflict - id: check-xml - id: check-yaml + args: ['--unsafe'] - id: debug-statements - id: end-of-file-fixer - id: requirements-txt-fixer diff --git a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md index dba6d4fe..818c7182 100644 --- a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md +++ b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md @@ -1,3 +1,44 @@ +--- +hide: + - toc +--- # Interval as a Basic Information Unit -Coming soon +Intervals serve as a fundamental unit of information by measuring the number of different +items, events, or symbols that occur between reseated in a sequence. + +The intervals for symbol `A` in the following sequence would be `[3, 3, 1, 1, 1, 2, 1]` +``` mermaid +block-beta + columns 12 + s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] + i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["1"]:1 i6["2"]:2 i7["1"]:1 +``` + +In general, a sequence does not necessarily end with the same symbol it begins with. +To cover all cases, we consider the sequence as a looped sequence representing an infinite pattern with the same characteristics as the original data +This cyclic approach corresponds to the idea of ​​representativeness heuristic. + +The intervals for symbol `C` in the following cycled sequence would be `[3, 9]` +``` mermaid +block-beta + columns 15 + s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] space s13["T"] s14["C"] + space i1["3"]:3 i2["9"]:10 + s12 --> s13 +``` + +The circular pattern preserves both the statistical properties and the order of elements. +Moreover, the average interval length is the inverse of the probability of an event, which directly relates intervals to probability. + +\begin{array}{|c|c|c|} +\hline + & \Delta_a & P \\ +\hline +A & \frac{3 + 3 + 1 + 1 + 1 + 2 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\ +\hline +C & \frac{3 + 9}{2} = \frac{12}{2} = 6 & \frac{2}{12} = \frac{1}{6} = 6^{-1} = \Delta_a^{-1} \\ +\hline +\end{array} + +This makes intervals a crucial informational unit that offers deeper insights into the sequence than individual occurrences alone. diff --git a/mkdocs.yml b/mkdocs.yml index 415f89e1..a96bf592 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -138,12 +138,16 @@ markdown_extensions: pygments_lang_class: true - pymdownx.inlinehilite - pymdownx.snippets - - pymdownx.superfences - tables - pymdownx.tabbed: alternate_style: true - pymdownx.arithmatex: generic: true + - pymdownx.superfences: + custom_fences: + - name: mermaid + class: mermaid + format: !!python/name:pymdownx.superfences.fence_code_format plugins: - autorefs From 130897accc5f6395cf49fb7b56565a0a85ec8f66 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 8 Jun 2025 00:26:49 +0200 Subject: [PATCH 10/17] Interval as information unit --- .../ideas/interval_as_a_basic_information_unit.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md index 818c7182..5e8b547f 100644 --- a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md +++ b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md @@ -7,12 +7,12 @@ hide: Intervals serve as a fundamental unit of information by measuring the number of different items, events, or symbols that occur between reseated in a sequence. -The intervals for symbol `A` in the following sequence would be `[3, 3, 1, 1, 1, 2, 1]` +The intervals for symbol `A` in the following sequence would be `[3, 3, 1, 1, 2, 1, 1]` ``` mermaid block-beta columns 12 s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] - i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["1"]:1 i6["2"]:2 i7["1"]:1 + i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["2"]:2 i6["1"]:1 i7["1"]:1 ``` In general, a sequence does not necessarily end with the same symbol it begins with. @@ -35,7 +35,7 @@ Moreover, the average interval length is the inverse of the probability of an ev \hline & \Delta_a & P \\ \hline -A & \frac{3 + 3 + 1 + 1 + 1 + 2 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\ +A & \frac{3 + 3 + 1 + 1 + 2 + 1 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\ \hline C & \frac{3 + 9}{2} = \frac{12}{2} = 6 & \frac{2}{12} = \frac{1}{6} = 6^{-1} = \Delta_a^{-1} \\ \hline From 70322344c56b281c6b52206ddba96250843c082c Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 8 Jun 2025 23:19:34 +0200 Subject: [PATCH 11/17] Finish ideas --- .../geometric_mean_based_characteristics.md | 33 ++++++++++++++++++- 1 file changed, 32 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md index 504d1552..988c11c2 100644 --- a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md +++ b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md @@ -1,3 +1,34 @@ +--- +hide: + - toc +--- # Geometric Mean as Alternative to Probability -Coming soon +At first glance, introducing the concept of [intervals](interval_as_a_basic_information_unit.md) may seem like an unnecessary complication. +After all, if the ultimate goal is to estimate the probability of a symbol, there are much simpler methods - counting the frequency of occurrence of a symbol relative to the total. + +However, this perspective begins to shift when we consider other types of aggregate functions beyond the arithmetic mean. +One particularly insightful example is the geometric mean of intervals. +While the arithmetic mean smooths out the "structure" of the data and bring us back to probability +(since the average interval between identical symbols is simply the inverse of their probability), +the geometric mean responds to the diversity of intervals in a fundamentally different way. + +If the intervals between repeated elements are uniform, the geometric mean and the arithmetic mean will be the same. +But as the intervals become more irregular — because symbols appear in bursts or clusters — [the geometric mean begins +to diverge from the arithmetic mean](https://en.wikipedia.org/wiki/AM%E2%80%93GM_inequality). +This makes it a sensitive indicator of the order within the sequence, not just the frequency. + +![AM-GM inequality visual proof](https://upload.wikimedia.org/wikipedia/commons/d/d9/AM_GM_inequality_visual_proof.svg) + +*Visual proof of the arithmetic mean - geometric mean inequality. Source: [wikipedia.org](https://en.wikipedia.org/wiki/File:AM_GM_inequality_visual_proof.svg)* + +Building on this idea, Former Order Analysis explored the potential of reinterpreting classical probabilistic and information-theoretic measures in terms of these intervals. +Instead of relying solely on symbol frequencies, it reformulated the measures using the geometric mean instead of probability (arithmetic mean). + +\begin{array}{|c|c|} +H= - \sum_{i=1}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{(n_j * \log_2 \sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})} & \\ +\end{array} + +These measures are fine-grained and sensitive to the temporal or spatial order of elements in a sequence. +Allows us to distinguish between sequences of symbols that may have identical probability distributions +but differ in the way those symbols are arranged - insight that traditional measures completely miss. From 487b3e5f457f70bfcb5cccdfc0e43182d227fd05 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sun, 8 Jun 2025 23:42:57 +0200 Subject: [PATCH 12/17] Finish ideas --- docs/fundamentals/ideas/index.md | 19 ++++++++++++++++++- 1 file changed, 18 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/index.md b/docs/fundamentals/ideas/index.md index 8c15c86d..b7aecd8b 100644 --- a/docs/fundamentals/ideas/index.md +++ b/docs/fundamentals/ideas/index.md @@ -1,3 +1,20 @@ # Ideas -Coming soon +Formal Order Analsis is based on five main ideas: + +* [Sequence as a Whole Object](sequence_as_a_whole_object.md) - +Treat a symbolic sequence as a unique whole, focusing on its internal structure and emergent properties. + +* [Order as a Property](order_as_a_property.md) - +By replacing each symbol in a sequence with its index in a dynamic alphabet, we define an "Order" — a new property that separates the content of the sequence from its composition. + +* [Congeneric Decomposition](congeneric_decomposition.md) - +This method breaks a symbolic sequence into layered sub-sequences where each layer isolates a single symbol’s positions, +preserving the sequence’s internal order and making its structure more analyzable. + +* [Interval as a Basic Information Unit](interval_as_a_basic_information_unit.md) - +Intervals measure the distance between repeated elements, revealing hidden patterns in repetition and spacing. + +* [Geometric Mean as Alternative to Probability](geometric_mean_based_characteristics.md) - +Using measurements based on the geometric mean of intervals instead of the probability of symbols (the arithmetic mean of occurrence) +allows us to move from recording simple frequency to recording the underlying structure and regularity in the arrangement of symbols. From 9b13402242476fbe18a67a72f1d0c8623b94b17f Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Mon, 9 Jun 2025 11:05:23 +0200 Subject: [PATCH 13/17] Fix formulas --- .../ideas/geometric_mean_based_characteristics.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md index 988c11c2..9c4d9bb5 100644 --- a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md +++ b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md @@ -26,9 +26,15 @@ Building on this idea, Former Order Analysis explored the potential of reinterpr Instead of relying solely on symbol frequencies, it reformulated the measures using the geometric mean instead of probability (arithmetic mean). \begin{array}{|c|c|} -H= - \sum_{i=1}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{(n_j * \log_2 \sum_{i=1}^{n_j} \frac{\Delta_{ij}}{n_j})} & \\ +\hline +Entropy & Average \ remoteness \\ +\hline +H= - \sum_{i=j}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{n_j \log_2 \Delta_{a_j}} & g = \frac{1}{n} * \sum_{i=j}^{m}{n_j \log_2{\Delta_{g_j}}} = \frac{1}{n} * \sum_{i=j}^{m}{\sum_{i=1}^{n_j} \log_2 \Delta_{ij}} \\ +\hline \end{array} +*Example of Shennon's Entropy analog - Average remoteness. Where $n$ - seqeunce length, $m$ - alphabet power, $n_j$ - count of element $j$-th, $\Delta_{a_j}$ - average mean of intervals for element $j$-th, $\Delta_{g_j}$ - geometric mean of intervals for element $j$-th, $\Delta_{ij}$ - $i$-th interval for element $j$-thg* + These measures are fine-grained and sensitive to the temporal or spatial order of elements in a sequence. Allows us to distinguish between sequences of symbols that may have identical probability distributions but differ in the way those symbols are arranged - insight that traditional measures completely miss. From cbaeeb8088b10cf63fc47bdfb3e72988b69b0185 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Mon, 9 Jun 2025 11:34:06 +0200 Subject: [PATCH 14/17] Fix typo --- docs/fundamentals/ideas/geometric_mean_based_characteristics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md index 9c4d9bb5..86ba5fb5 100644 --- a/docs/fundamentals/ideas/geometric_mean_based_characteristics.md +++ b/docs/fundamentals/ideas/geometric_mean_based_characteristics.md @@ -29,7 +29,7 @@ Instead of relying solely on symbol frequencies, it reformulated the measures us \hline Entropy & Average \ remoteness \\ \hline -H= - \sum_{i=j}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{n_j \log_2 \Delta_{a_j}} & g = \frac{1}{n} * \sum_{i=j}^{m}{n_j \log_2{\Delta_{g_j}}} = \frac{1}{n} * \sum_{i=j}^{m}{\sum_{i=1}^{n_j} \log_2 \Delta_{ij}} \\ +H= - \sum_{j=1}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{n_j \log_2 \Delta_{a_j}} & g = \frac{1}{n} * \sum_{j=1}^{m}{n_j \log_2{\Delta_{g_j}}} = \frac{1}{n} * \sum_{j=1}^{m}{\sum_{i=1}^{n_j} \log_2 \Delta_{ij}} \\ \hline \end{array} From 21ecb64121b59420ba8985f8995fd55252c33c07 Mon Sep 17 00:00:00 2001 From: Nikolay Pozdnichenko Date: Mon, 9 Jun 2025 23:32:55 +0600 Subject: [PATCH 15/17] Update order_as_a_property.md Added references section --- docs/fundamentals/ideas/order_as_a_property.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md index 5de14681..32b2851f 100644 --- a/docs/fundamentals/ideas/order_as_a_property.md +++ b/docs/fundamentals/ideas/order_as_a_property.md @@ -7,7 +7,7 @@ hide: Formal order analysis defines a special property of symbolic sequences - an Order. The order is a sequence of natural numbers obtained from the original symbolic sequence by replacing each of its elements with a natural number corresponding to the index of this element in the alphabet -sorted by the appearance of the elements in the original sequence. +sorted by the appearance of the elements in the original sequence [1, 2, 3]. The concept of an Order can be conveniently demonstrated using an example: @@ -74,3 +74,9 @@ print("Order =", orderA) text-align: center !important; } + +# References: + +1. Curtis Cooper and Robert E. Kennedy. 1992. Patterns, automata, and Stirling numbers of the second kind. Math. Comput. Educ. 26, 2 (Spring 1992), 120–124. +2. Gumenjuk A., Kostyshin A., Simonova S. An approach to the research of the structure of linguistic and musical texts. Glottometrics. 2002. № 3. P. 61–89. +3. (In russian) V.I. Arnold, Complexity of finite sequences of zeros and ones and geometry of finite function spaces: el. print, 2005. http://mms.mathnet.ru/meetings/2005/arnold.pdf From 4fa971919cd6c6909c55cc0430ddd1707705f081 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sat, 14 Jun 2025 20:00:03 +0200 Subject: [PATCH 16/17] Added marmaid diagram --- docs/assets/js/mathjax.js | 4 +- .../ideas/congeneric_decomposition.md | 206 +++++++++++-- .../fundamentals/ideas/order_as_a_property.md | 284 +++++++++++++++++- 3 files changed, 454 insertions(+), 40 deletions(-) diff --git a/docs/assets/js/mathjax.js b/docs/assets/js/mathjax.js index 0c7803cf..561944cd 100644 --- a/docs/assets/js/mathjax.js +++ b/docs/assets/js/mathjax.js @@ -1,9 +1,11 @@ window.MathJax = { + loader: {load: ['[tex]/colortbl']}, tex: { inlineMath: [["\\(", "\\)"]], displayMath: [["\\[", "\\]"]], processEscapes: true, - processEnvironments: true + processEnvironments: true, + packages: {'[+]': ['colortbl']}, }, options: { ignoreHtmlClass: ".*|", diff --git a/docs/fundamentals/ideas/congeneric_decomposition.md b/docs/fundamentals/ideas/congeneric_decomposition.md index bedc9037..6ce33b9a 100644 --- a/docs/fundamentals/ideas/congeneric_decomposition.md +++ b/docs/fundamentals/ideas/congeneric_decomposition.md @@ -11,35 +11,191 @@ This reversible process preserves the order of the sequence and allows the origi The concept of Cogeneric decomposition can be demonstrated using an example: -Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE` congeneric decomposition +Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT` congeneric decomposition could be presented by the following table, where each row is a congeneric sequence and `-` is an empty position in a congeneric sequence. -| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | -|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| -| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | -| I | - | - | - | - | - | I | - | - | - | - | - | - | I | - | - | - | - | - | - | - | - | I | - | I | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | N | - | - | - | - | - | - | - | N | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | N | - | - | -| - | - | T | - | - | - | - | - | - | - | - | - | - | - | - | - | T | - | - | - | - | - | - | - | - | T | - | - | T | - | - | - | - | - | - | T | - | T | - | - | - | - | - | - | - | - | -| - | - | - | E | - | - | - | - | E | - | - | E | - | - | - | - | - | - | E | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | E | -| - | - | - | - | L | L | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | L | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | G | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | G | - | -| - | - | - | - | - | - | - | - | - | - | C | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | C | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - |    | - | - |    | - | - | - |    | - | - | - | - | - | - | - |    | - | - |    | - | - | - | - | - |    | - | - |    | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | S | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | H | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | H | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | A | - | - | - | - | - | - | - | - | - | - | A | - | A | - | - | - | - | - | - | - | - | A | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | B | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | Y | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | O | - | - | - | - | - | - | - | - | O | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | D | - | - | - | - | - | - | - | - | - | - | - | - | - | -| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | P | - | - | - | - | - | - | - | - | - | - | - | - - -Congeneric sequence for `I` + + + + + +``` mermaid +block-beta + columns 36 + seq1["I"] seq2["N"] seq3["T"] seq4["E"] seq5["L"] seq6["L"] seq7["I"] seq8["G"] seq9["E"] seq10["N"] + seq11["C"] seq12["E"] seq13[" "] seq14["I"] seq15["S"] seq16[" "] seq17["T"] seq18["H"] seq19["E"] seq20[" "] + seq21["A"] seq22["B"] seq23["I"] seq24["L"] seq25["I"] seq26["T"] seq27["Y"] seq28[" "] seq29["T"] seq30["O"] + seq31[" "] seq32["A"] seq33["D"] seq34["A"] seq35["P"] seq36["T"] + + space:36 + + i1["I"] i2["-"] i3["-"] i4["-"] i5["-"] i6["-"] i7["I"] i8["-"] i9["-"] i10["-"] + i11["-"] i12["-"] i13["-"] i14["I"] i15["-"] i16["-"] i17["-"] i18["-"] i19["-"] i20["-"] + i21["-"] i22["-"] i23["I"] i24["-"] i25["I"] i26["-"] i27["-"] i28["-"] i29["-"] i30["-"] + i31["-"] i32["-"] i33["-"] i34["-"] i35["-"] i36["-"] + + n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"] + n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"] + n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"] n30["-"] + n31["-"] n32["-"] n33["-"] n34["-"] n35["-"] n36["-"] + + t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"] + t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"] + t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"] t30["-"] + t31["-"] t32["-"] t33["-"] t34["-"] t35["-"] t36["T"] + + e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"] + e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"] + e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"] + e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"] + + l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"] + l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"] + l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"] l30["-"] + l31["-"] l32["-"] l33["-"] l34["-"] l35["-"] l36["-"] + + g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"] + g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"] + g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"] g30["-"] + g31["-"] g32["-"] g33["-"] g34["-"] g35["-"] g36["-"] + + c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"] + c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"] + c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"] c30["-"] + c31["-"] c32["-"] c33["-"] c34["-"] c35["-"] c36["-"] + + sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"] + sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "] + sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28[" "] sp29["-"] sp30["-"] + sp31[" "] sp32["-"] sp33["-"] sp34["-"] sp35["-"] sp36["-"] + + s1["-"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"] + s11["-"] s12["-"] s13["-"] s14["-"] s15["S"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"] + s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"] s30["-"] + s31["-"] s32["-"] s33["-"] s34["-"] s35["-"] s36["-"] + + h1["-"] h2["-"] h3["-"] h4["-"] h5["-"] h6["-"] h7["-"] h8["-"] h9["-"] h10["-"] + h11["-"] h12["-"] h13["-"] h14["-"] h15["-"] h16["-"] h17["-"] h18["H"] h19["-"] h20["-"] + h21["-"] h22["-"] h23["-"] h24["-"] h25["-"] h26["-"] h27["-"] h28["-"] h29["-"] h30["-"] + h31["-"] h32["-"] h33["-"] h34["-"] h35["-"] h36["-"] + + a1["-"] a2["-"] a3["-"] a4["-"] a5["-"] a6["-"] a7["-"] a8["-"] a9["-"] a10["-"] + a11["-"] a12["-"] a13["-"] a14["-"] a15["-"] a16["-"] a17["-"] a18["-"] a19["-"] a20["-"] + a21["A"] a22["-"] a23["-"] a24["-"] a25["-"] a26["-"] a27["-"] a28["-"] a29["-"] a30["-"] + a31["-"] a32["A"] a33["-"] a34["A"] a35["-"] a36["-"] + + b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"] + b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"] + b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"] b30["-"] + b31["-"] b32["-"] b33["-"] b34["-"] b35["-"] b36["-"] + + y1["-"] y2["-"] y3["-"] y4["-"] y5["-"] y6["-"] y7["-"] y8["-"] y9["-"] y10["-"] + y11["-"] y12["-"] y13["-"] y14["-"] y15["-"] y16["-"] y17["-"] y18["-"] y19["-"] y20["-"] + y21["-"] y22["-"] y23["-"] y24["-"] y25["-"] y26["-"] y27["Y"] y28["-"] y29["-"] y30["-"] + y31["-"] y32["-"] y33["-"] y34["-"] y35["-"] y36["-"] + + o1["-"] o2["-"] o3["-"] o4["-"] o5["-"] o6["-"] o7["-"] o8["-"] o9["-"] o10["-"] + o11["-"] o12["-"] o13["-"] o14["-"] o15["-"] o16["-"] o17["-"] o18["-"] o19["-"] o20["-"] + o21["-"] o22["-"] o23["-"] o24["-"] o25["-"] o26["-"] o27["-"] o28["-"] o29["-"] o30["O"] + o31["-"] o32["-"] o33["-"] o34["-"] o35["-"] o36["-"] + + d1["-"] d2["-"] d3["-"] d4["-"] d5["-"] d6["-"] d7["-"] d8["-"] d9["-"] d10["-"] + d11["-"] d12["-"] d13["-"] d14["-"] d15["-"] d16["-"] d17["-"] d18["-"] d19["-"] d20["-"] + d21["-"] d22["-"] d23["-"] d24["-"] d25["-"] d26["-"] d27["-"] d28["-"] d29["-"] d30["-"] + d31["-"] d32["-"] d33["D"] d34["-"] d35["-"] d36["-"] + + p1["-"] p2["-"] p3["-"] p4["-"] p5["-"] p6["-"] p7["-"] p8["-"] p9["-"] p10["-"] + p11["-"] p12["-"] p13["-"] p14["-"] p15["-"] p16["-"] p17["-"] p18["-"] p19["-"] p20["-"] + p21["-"] p22["-"] p23["-"] p24["-"] p25["-"] p26["-"] p27["-"] p28["-"] p29["-"] p30["-"] + p31["-"] p32["-"] p33["-"] p34["-"] p35["P"] p36["-"] + + classDef c1 fill:#ff7f0e,color:#fff; + classDef c2 fill:#ffbb78,color:#000; + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + classDef c5 fill:#d62728,color:#fff; + classDef c6 fill:#ff9896,color:#000; + classDef c7 fill:#9467bd,color:#fff; + classDef c8 fill:#c5b0d5,color:#000; + classDef c9 fill:#8c564b,color:#fff; + classDef c10 fill:#c49c94,color:#000; + classDef c11 fill:#e377c2,color:#fff; + classDef c12 fill:#f7b6d2,color:#000; + classDef c13 fill:#bcbd22,color:#fff; + classDef c14 fill:#dbdb8d,color:#000; + classDef c15 fill:#17becf,color:#fff; + classDef c16 fill:#9edae5,color:#000; + + class seq1,seq7,seq14,seq23,seq25,i1,i7,i14,i23,i25 c1 + class seq2,seq10,n2,n10 c2 + class seq3,seq17,seq26,seq29,seq36,t3,t17,t26,t29,t36 c3 + class seq4,seq9,seq12,seq19,e4,e9,e12,e19 c4 + class seq5,seq6,seq24,l5,l6,l24 c5 + class seq8,g8 c6 + class seq11,c11 c7 + class seq13,seq16,seq20,seq28,seq31,sp13,sp16,sp20,sp28,sp31 c8 + class seq15,s15 c9 + class seq18,h18 c10 + class seq21,seq32,seq34,a21,a32,a34 c11 + class seq22,b22 c12 + class seq27,y27 c13 + class seq30,o30 c14 + class seq33,d33 c15 + class seq35,p35 c16 ``` -I - - - - - I - - - - - - I - - - - - - - - I - I - - - - - - - - - - - - - - - - - - - - - + +Congeneric sequence for `E` +``` mermaid +block-beta + columns 36 + e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"] + e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"] + e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"] + e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"] ``` -could be a part of multiple symbol sequences that have the same order of `I` element. + + +could be a part of multiple symbol sequences that have the same order of `E` element. While keeping the main idea, the congeneric decomposition could be applied, with a flavor, to any type of special case symbolic sequences, such as Order. diff --git a/docs/fundamentals/ideas/order_as_a_property.md b/docs/fundamentals/ideas/order_as_a_property.md index 32b2851f..401d2928 100644 --- a/docs/fundamentals/ideas/order_as_a_property.md +++ b/docs/fundamentals/ideas/order_as_a_property.md @@ -12,35 +12,291 @@ sorted by the appearance of the elements in the original sequence [1, 2, 3]. The concept of an Order can be conveniently demonstrated using an example: -Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE` +Let's assume there is a symbolic sequence + +``` mermaid +block-beta + columns 36 + s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"] + s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "] + s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"] + s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"] +``` + Find and enumirate the first appearance of each element -| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | -|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| -| 1 | 2 | 3 | 4 | 5 | | | 6 | | | 7 | | 8 | | 9 | | | 10 | | | 11 | 12 | | | | | 13 | | | 14 | | | 15 | | 16 | | | | | | | | | | | | +``` mermaid +block-beta + columns 36 + s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"] + s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "] + s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"] + s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"] + + space:36 + + i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] space:2 i6_1["6"] space:2 i7_1["7"] space + i8_1["8"] space i9_1["9"] space:2 i10_1["10"] space:2 + i11_1["11"] i12_1["12"] space:4 i13_1["13"] + space:2 i14_1["14"] space:2 + i15_1["15"] space i16_1["16"] + + classDef c1 fill:#ff7f0e,color:#fff; + classDef c2 fill:#ffbb78,color:#000; + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + classDef c5 fill:#d62728,color:#fff; + classDef c6 fill:#ff9896,color:#000; + classDef c7 fill:#9467bd,color:#fff; + classDef c8 fill:#c5b0d5,color:#000; + classDef c9 fill:#8c564b,color:#fff; + classDef c10 fill:#c49c94,color:#000; + classDef c11 fill:#e377c2,color:#fff; + classDef c12 fill:#f7b6d2,color:#000; + classDef c13 fill:#bcbd22,color:#fff; + classDef c14 fill:#dbdb8d,color:#000; + classDef c15 fill:#17becf,color:#fff; + classDef c16 fill:#9edae5,color:#000; + + class s1,i1_1 c1 + class s2,i2_1 c2 + class s3,i3_1 c3 + class s4,i4_1 c4 + class s5,i5_1 c5 + class s8,i6_1 c6 + class s11,i7_1 c7 + class s13,i8_1 c8 + class s15,i9_1 c9 + class s18,i10_1 c10 + class s21,i11_1 c11 + class s22,i12_1 c12 + class s27,i13_1 c13 + class s30,i14_1 c14 + class s33,i15_1 c15 + class s35,i16_1 c16 + + s1 --> i1_1 + s2 --> i2_1 + s3 --> i3_1 + s4 --> i4_1 + s5 --> i5_1 + s8 --> i6_1 + s11 --> i7_1 + s13 --> i8_1 + s15 --> i9_1 + s18 --> i10_1 + s21 --> i11_1 + s22 --> i12_1 + s27 --> i13_1 + s30 --> i14_1 + s33 --> i15_1 + s35 --> i16_1 +``` -The alphabet for the sequence would be sequence of unique elements: +The alphabet (with indexes) for the sequence would be sequence of unique elements: -| I | N | T | E | L | G | C | | S | H | A | B | Y | O | D | P | -|---|---|---|---|---|---|---|---|---|----|----|----|----|----|----|----| -| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | + +``` mermaid +block-beta + columns 16 + s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s8["G"] + s11["C"] s13[" "] s15["S"] s18["H"] + s21["A"] s22["B"] s27["Y"] s30["O"] + s33["D"] s35["P"] + + space:16 + + i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i6_1["6"] i7_1["7"] + i8_1["8"] i9_1["9"] i10_1["10"] + i11_1["11"] i12_1["12"] i13_1["13"] + i14_1["14"] + i15_1["15"] i16_1["16"] + + classDef c1 fill:#ff7f0e,color:#fff; + classDef c2 fill:#ffbb78,color:#000; + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + classDef c5 fill:#d62728,color:#fff; + classDef c6 fill:#ff9896,color:#000; + classDef c7 fill:#9467bd,color:#fff; + classDef c8 fill:#c5b0d5,color:#000; + classDef c9 fill:#8c564b,color:#fff; + classDef c10 fill:#c49c94,color:#000; + classDef c11 fill:#e377c2,color:#fff; + classDef c12 fill:#f7b6d2,color:#000; + classDef c13 fill:#bcbd22,color:#fff; + classDef c14 fill:#dbdb8d,color:#000; + classDef c15 fill:#17becf,color:#fff; + classDef c16 fill:#9edae5,color:#000; + + class s1,i1_1 c1 + class s2,i2_1 c2 + class s3,i3_1 c3 + class s4,i4_1 c4 + class s5,i5_1 c5 + class s8,i6_1 c6 + class s11,i7_1 c7 + class s13,i8_1 c8 + class s15,i9_1 c9 + class s18,i10_1 c10 + class s21,i11_1 c11 + class s22,i12_1 c12 + class s27,i13_1 c13 + class s30,i14_1 c14 + class s33,i15_1 c15 + class s35,i16_1 c16 + + s1 --> i1_1 + s2 --> i2_1 + s3 --> i3_1 + s4 --> i4_1 + s5 --> i5_1 + s8 --> i6_1 + s11 --> i7_1 + s13 --> i8_1 + s15 --> i9_1 + s18 --> i10_1 + s21 --> i11_1 + s22 --> i12_1 + s27 --> i13_1 + s30 --> i14_1 + s33 --> i15_1 + s35 --> i16_1 + + i1_1 --> s1 + i2_1 --> s2 + i3_1 --> s3 + i4_1 --> s4 + i5_1 --> s5 + i6_1 --> s8 + i7_1 --> s11 + i8_1 --> s13 + i9_1 --> s15 + i10_1 --> s18 + i11_1 --> s21 + i12_1 --> s22 + i13_1 --> s27 + i14_1 --> s30 + i15_1 --> s33 + i16_1 --> s35 +``` Determine the order of the sequence by replacing each element of the sequence with its corresponding alphabet index +``` mermaid +block-beta + columns 36 + s1["I"] s2["N"] s3["T"] s4["E"] s5["L"] s6["L"] s7["I"] s8["G"] s9["E"] s10["N"] + s11["C"] s12["E"] s13[" "] s14["I"] s15["S"] s16[" "] s17["T"] s18["H"] s19["E"] s20[" "] + s21["A"] s22["B"] s23["I"] s24["L"] s25["I"] s26["T"] s27["Y"] s28[" "] s29["T"] s30["O"] + s31[" "] s32["A"] s33["D"] s34["A"] s35["P"] s36["T"] -| I | N | T | E | L | L | I | G | E | N | C | E |    | I | S |    | T | H | E |    | A | B | I | L | I | T | Y |    | T | O |    | A | D | A | P | T |    | T | O |    | C | H | A | N | G | E | -|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| -| 1 | 2 | 3 | 4 | 5 | 5 | 1 | 6 | 4 | 2 | 7 | 4 | 8 | 1 | 9 | 8 | 3 | 10 | 4 | 8 | 11 | 12 | 1 | 5 | 1 | 3 | 13 | 8 | 3 | 14 | 8 | 11 | 15 | 11 | 16 | 3 | 8 | 3 | 14 | 8 | 7 | 10 | 11 | 2 | 6 | 4 | + space:36 -The order of symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT TO CHANGE` is + i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i5_2["5"] i1_2["1"] i6_1["6"] i4_2["4"] i2_2["2"] i7_1["7"] i4_3["4"] + i8_1["8"] i1_3["1"] i9_1["9"] i8_2["8"] i3_2["3"] i10_1["10"] i4_4["4"] i8_3["8"] + i11_1["11"] i12_1["12"] i1_4["1"] i5_3["5"] i1_5["1"] i3_3["3"] i13_1["13"] + i8_4["8"] i3_4["3"] i14_1["14"] i8_5["8"] i11_2["11"] + i15_1["15"] i11_3["11"] i16_1["16"] i3_5["3"] -| 1 | 2 | 3 | 4 | 5 | 5 | 1 | 6 | 4 | 2 | 7 | 4 | 8 | 1 | 9 | 8 | 3 | 10 | 4 | 8 | 11 | 12 | 1 | 5 | 1 | 3 | 13 | 8 | 3 | 14 | 8 | 11 | 15 | 11 | 16 | 3 | 8 | 3 | 14 | 8 | 7 | 10 | 11 | 2 | 6 | 4 | -|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| + classDef c1 fill:#ff7f0e,color:#fff; + classDef c2 fill:#ffbb78,color:#000; + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + classDef c5 fill:#d62728,color:#fff; + classDef c6 fill:#ff9896,color:#000; + classDef c7 fill:#9467bd,color:#fff; + classDef c8 fill:#c5b0d5,color:#000; + classDef c9 fill:#8c564b,color:#fff; + classDef c10 fill:#c49c94,color:#000; + classDef c11 fill:#e377c2,color:#fff; + classDef c12 fill:#f7b6d2,color:#000; + classDef c13 fill:#bcbd22,color:#fff; + classDef c14 fill:#dbdb8d,color:#000; + classDef c15 fill:#17becf,color:#fff; + classDef c16 fill:#9edae5,color:#000; + class s1,s7,s14,s23,s25,i1_1,i1_2,i1_3,i1_4,i1_5 c1 + class s2,s10,i2_1,i2_2 c2 + class s3,s17,s26,s29,s36,i3_1,i3_2,i3_3,i3_4,i3_5 c3 + class s4,s9,s12,s19,i4_1,i4_2,i4_3,i4_4 c4 + class s5,s6,s24,i5_1,i5_2,i5_3 c5 + class s8,i6_1 c6 + class s11,i7_1 c7 + class s13,s16,s20,s28,s31,i8_1,i8_2,i8_3,i8_4,i8_5 c8 + class s15,i9_1 c9 + class s18,i10_1 c10 + class s21,s32,s34,i11_1,i11_2,i11_3 c11 + class s22,i12_1 c12 + class s27,i13_1 c13 + class s30,i14_1 c14 + class s33,i15_1 c15 + class s35,i16_1 c16 + s1 --> i1_1 + s7 --> i1_2 + s14 --> i1_3 + s23 --> i1_4 + s25 --> i1_5 + + s2 --> i2_1 + s10 --> i2_2 + + s3 --> i3_1 + s17 --> i3_2 + s26 --> i3_3 + s29 --> i3_4 + s36 --> i3_5 + + s4 --> i4_1 + s9 --> i4_2 + s12 --> i4_3 + s19--> i4_4 + + s5 --> i5_1 + s6 --> i5_2 + s24 --> i5_3 + + s8 --> i6_1 + + s11 --> i7_1 + + s13 --> i8_1 + s16 --> i8_2 + s20 --> i8_3 + s28 --> i8_4 + s31 --> i8_5 + + s15 --> i9_1 + s18 --> i10_1 + + s21 --> i11_1 + s32 --> i11_2 + s34 --> i11_3 + + s22 --> i12_1 + s27 --> i13_1 + s30 --> i14_1 + s33 --> i15_1 + s35 --> i16_1 +``` + + +The order of symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT` is + +``` mermaid +block-beta + columns 36 + i1_1["1"] i2_1["2"] i3_1["3"] i4_1["4"] i5_1["5"] i5_2["5"] i1_2["1"] i6_1["6"] i4_2["4"] i2_2["2"] i7_1["7"] i4_3["4"] + i8_1["8"] i1_3["1"] i9_1["9"] i8_2["8"] i3_2["3"] i10_1["10"] i4_4["4"] i8_3["8"] + i11_1["11"] i12_1["12"] i1_4["1"] i5_3["5"] i1_5["1"] i3_3["3"] i13_1["13"] + i8_4["8"] i3_4["3"] i14_1["14"] i8_5["8"] i11_2["11"] + i15_1["15"] i11_3["11"] i16_1["16"] i3_5["3"] +``` + Despite the triviality of the concept Order, it allows us to separate the elements and composition of a sequence and to define the compositional equivalence of different sequences. Example of sequences with equals orders: From 419b3449ac4a33496e915b1d4548d46843c58c18 Mon Sep 17 00:00:00 2001 From: Igor Rodionov Date: Sat, 14 Jun 2025 20:07:40 +0200 Subject: [PATCH 17/17] Added intervals coloring --- .../ideas/interval_as_a_basic_information_unit.md | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md index 5e8b547f..634e1606 100644 --- a/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md +++ b/docs/fundamentals/ideas/interval_as_a_basic_information_unit.md @@ -13,6 +13,11 @@ block-beta columns 12 s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["2"]:2 i6["1"]:1 i7["1"]:1 + + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + class s1,s4,s7,s8,s9,s11,s12 c3 + class i1,i2,i3,i4,i5,i6,i7 c4 ``` In general, a sequence does not necessarily end with the same symbol it begins with. @@ -26,6 +31,11 @@ block-beta s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] space s13["T"] s14["C"] space i1["3"]:3 i2["9"]:10 s12 --> s13 + + classDef c3 fill:#2ca02c,color:#fff; + classDef c4 fill:#98df8a,color:#000; + class s2,s5,s14 c3 + class i1,i2 c4 ``` The circular pattern preserves both the statistical properties and the order of elements.