Happy Pi Day! Today (3/14) we celebrate the most famous mathematical constant: ๐น โ 3.141592653589793…
๐น is irrational and transcendental, appears in circles, waves, probability, physics, and random walks.
Wolfram Language (with its built-in ๐น constant, excellent rational support, symbolic programming, and vast collection of Number Theory functions) makes experimenting with ๐น especially easy and enjoyable.
In this document (notebook) we explore a selection of formulas and algorithms.
1. Continued fraction approximation
The built-in Wolfram Language (WL) constant Pi (or ฯ ) can be computed to an arbitrary high precision:
Let us compare it with a continued fraction approximation. For example, using the (first) sequence line of On-line Encyclopedia of Integer Sequences (OEIS) A001203 produces ฯ with precision 100:
Next, we show the Pareto principle manifestation of for the continued fraction terms. First we observe that the terms a distribution similar to Benford’s law :
This document (notebook) describes three ways of making mazes (or labyrinths) using graphs. The first two are based on rectangular grids; the third on a hexagonal grid.
All computational graph features discussed here are provided by the Graph functionalities of Wolfram Language.
TL;DR
Just see the maze pictures below. (And try to solve the mazes.)
Procedure outline
The first maze is made by a simple procedure which is actually some sort of cheating:
A regular rectangular grid graph is generated with random weights associated with its edges.
The (minimum) spanning tree for that graph is found.
That tree is plotted with exaggeratedly large vertices and edges, so the graph plot looks like a maze.
This is “the cheat” — the maze walls are not given by the graph.
The second maze is made “properly”:
Two interlacing regular rectangular grid graphs are created.
The second one has one less row and one less column than the first.
The vertex coordinates of the second graph are at the centers of the rectangles of the first graph.
The first graph provides the maze walls; the second graph is used to make paths through the maze.
In other words, to create a solvable maze.
Again, random weights are assigned to edges of the second graph, and a minimum spanning tree is found.
There is a convenient formula that allows using the spanning tree edges to remove edges from the first graph.
In that way, a proper maze is derived.
The third maze is again made “properly” using the procedure above with two modifications:
Two interlacing regular grid graphs are created: one over a hexagonal grid, the other over a triangular grid.
The hexagonal grid graph provides the maze walls; the triangular grid graph provides the maze paths.
Since the formula for wall removal is hard to derive, a more robust and universal method based on nearest neighbors is used.
Simple Maze
In this section, we create a simple, “cheating” maze.
Remark: The steps are easy to follow, given the procedure outlined in the introduction.
The “maze” above looks like a maze because the vertices and edges are rectangular with matching sizes, and they are thicker than the spaces between them. In other words, we are cheating.
To make that cheating construction clearer, let us plot the shortest path from the bottom left to the top right and color the edges in pink (salmon) and the vertices in red:
finder = Nearest[Thread[GraphEmbedding[g1] -> VertexList[g1]]]
Take a maze edge and its vertex points:
e = First@EdgeList[mazePath];
aMazePathCoords = Association@Thread[VertexList[mazePath] -> GraphEmbedding[mazePath]];
points = List @@ (e /. aMazePathCoords)
Find the edge’s midpoint and the nearest wall-graph vertices:
This document (notebook) discusses number theory properties and relationships of the integer 2026.
The integer 2026 is semiprime and a happy number, with 365 as one of its primitive roots. Although 2026 may not be particularly noteworthy in number theory, this provides a great excuse to create various elaborate visualizations that reveal some interesting aspects of the number.
A happy number is a number for which iteratively summing the squares of its digits eventually reaches 1 (e.g., 13 -> 10 -> 1). Here is a check that 2026 is happy:
ResourceFunction["HappyNumberQ"][2026]
Out[]= True
Here is the corresponding trail of digit-square sums:
The decomposition of $2026$ as $2 * 1013$ means the multiplicative group modulo $2026$ has primitive roots. A primitive root exists for an integer $n$ if and only if $n$ is $1$, $2$,$4$, $p^k$ , or $2 p^k$ , where $k$ is an odd prime and $k>0$ .
We can check additional facts about 2026, such as whether it is “square-free” , among other properties. However, let us compare these with the feature-rich 2025 in the next section.
Comparison with 2025
Here is a side-by-side comparison of key number theory properties for 2025 and 2026.
Property
2025
2026
Notes
Prime or Composite
Composite
Composite
Both non-prime.
Prime Factorization
3^4 * 5^2 (81 * 25)
2 * 1013
2025 has repeated small primes; 2026 is a semiprime (product of two distinct primes).
Number of Divisors
15 (highly composite for its size)
4 (1, 2, 1013, 2026)
2025 has many divisors; 2026 has very few.
Perfect Square
Yes (45^2 = 2025)
No
Major highlight for 2025—rare square year.
Sum of Cubes
Yes (1^3 + 2^3 + … + 9^3 = (1 + … + 9)^2 = 2025)
No
Iconic property for 2025 (Nicomachus’s theorem).
Happy Number
No (process leads to cycle including 4)
Yes (repeated squared digit sums reach 1)
Key point for 2026—its main “happy” trait.
Harshad Number
Yes (divisible by 9)
No (not divisible by 10)
2025 qualifies; 2026 does not.
Primitive Roots
No
Yes
This is a relatively rare property to have.
Other Notable Traits
{(20 + 25)^2 = 2025, Sum of first 45 odd numbers, Deficient number, Many pattern-based representations}
{Even number, Deficient number, Few special patterns}
2025 is packed with elegant properties; 2026 is more “plain” beyond being happy.
Overall “Interest” Level
Highly interesting—celebrated in math communities for squares, cubes, and patterns
Relatively uninteresting—basic semiprime with no standout geometric or sum properties
Reinforces blog’s angle.
To summarize:
2025ย stands out as a mathematically rich number, often highlighted in puzzles and articles for its perfect square status and connections to sums of cubes and odd numbers.
2026ย , in contrast, has fewer flashy properties — it’s a straightforward even semiprime — but it qualifies as aย happy numberย and it has a primitive root.
Here is a computationally generated comparison table of most of the properties found in the table above:
Let us create and plot a graph showing the trails of all happy numbers less than or equal to 2026. Below, we identify these numbers and their corresponding trails:
There is a theorem by Gauss stating that any integer can be represented as a sum of at most three triangular numbers. Here we find an “interesting” solution:
sol = FindInstance[{2026 == PolygonalNumber[i] + PolygonalNumber[j] + PolygonalNumber[k], i > 10, j > 10, k > 10}, {i, j, k}, Integers]
Remark: It is interesting that 365 (the number of days in a common calendar year) is a primitive root of the multiplicative group generated by 2026 . Not many years have this property this century; many do not have primitive roots at all.
Here we find the position of 2026 in that sequence:
Position[seq, 2026]
Out[]= {{18}}
Given the title of the sequence and the extracted position of $2026$ , this means that the number of disconnected 4-regular graphs with 17 vertices is $2026$. ($17$ because the sequence offset is $0$.) Let us create a few graphs from that set by using the 5-vertex complete graph $\left(K_5\right.$) and circulant graphs . Here is an example of such a graph:
Number theory provides many opportunities for visualizations, so I included utilities for some of the popular patterns in “Math::NumberTheory”, [AAp1] andย “NumberTheoryUtilities”.
The number of years in this century that have primitive roots and have 365 as a primitive root is less than the number of years that are happy numbers.
I would say I spent too much time finding a good, suitable, Christmas-themed combination of colors for the trails graph.
This document (notebook) discusses different combinations of Grammar-Based Parser-Interpreters (GBPI) and Large Language Models (LLMs) to generate executable code from Natural Language Computational Specifications (NLCM). We have theย softย assumption that the NLCS adhere to a certain relatively small Domain Specific Language (DSL) or use terminology from that DSL. Another assumption is that the target software packages are not necessarily well-known by the LLMs, i.e. direct LLM requests for code using them would produce meaningless results.
We want to do such combinations because:
GBPI are fast, precise, but with a narrow DSL scope
LLMs can be unreliable and slow, but with a wide DSL scope
Because of GBPI and LLMs are complementary technologies with similar and overlapping goals the possible combinations are many. We concentrate on two of the most straightforward designs: (1) judged parallel race of methods execution, and (2) using LLMs as a fallback method if grammar parsing fails. We show asynchronous programmingimplementations for both designs using the Wolfram Language function LLMGraph.
The rest of the document is structured as follows:
Initial grammar-LLM combinations
Assumptions, straightforward designs, and trade-offs
Comprehensive combinations enumeration (attempt)
Tabular and morphological analysis breakdown
Three methods for parsing ML DSL specs into Raku code
One grammar-based, two LLM-based
Parallel execution with an LLM judge
Straightforward, but computationally wasteful and expensive
Grammar-to-LLM fallback mechanism
The easiest and most robust solution
Concluding comments and observations
TL;DR
Combining grammars and LLMs produces robust translators.
Three translators with different faithfulness and coverage are demonstrated and used.
Two of the simplest, yet effective, combinations are implemented and demonstrated.
Parallel race and grammar-to-LLM fallback.
Asynchronous implementations with LLM-graphs are a very good fit!
Just look at the LLM-graph plots (and be done reading.)
Initial Combinations and Associated Assumptions
The goal is to combine grammar-based parser-interpreters with LLMs in order to achieve robust parsing and interpretation of computational workflow specifications.
Here are some example combinations of these approaches:
A few methods, both grammar-based and LLM-based, are initiated in parallel. Whichever method produces a correct result first is selected as the answer.
This approach assumes that when the grammar-based methods are effective, they will finish more quickly than the LLM-based methods.
The grammar method is invoked first; if it fails, an LLM method (or a sequence of LLM methods) is employed.
LLMs are utilized at the grammar-rule level to provide matching objects that the grammar can work with.
If the grammar method fails, an LLM normalizer for user commands is invoked to generate specifications that the grammar can parse.
It is important to distinguish between declarative specifications and those that prescribe specific steps.
For a workflow given as a list of steps the grammar parser may successfully parse most steps, but LLMs may be required for a few exceptions.
The main trade-off in these approaches is as follows:
Grammar methods are challenging to develop but can be very fast and precise.
Precision can be guaranteed and rigorously tested.
LLM methods are quicker to develop but tend to be slower and can be unreliable, particularly for less popular workflows, programming languages, and packages.
Also, combinations based on LLM tools (aka LLM external function calling) are not considered because LLM-tools invocation is too unpredictable and unreliable.
Comprehensive breakdown (attempt)
This section has a “concise” table that expands the combinations list above into the main combinatorial strategies for Grammar *** LLMs** for robust parsing and interpretation of workflow specifications. The table is not an exhaustive list of such combinations, but illustrates their diversity and, hopefully, can give ideas for future developments.
A few summary points (for table’s content/subject):
The most robust systems combineย grammar precisionย withย LLM adaptabilityย , typically by putting grammars first and using LLMs for repair, normalization, expansions, or semantic interpretation (i.e. “fallback”.)
Table: Combination Patterns for Parsing Workflow Specifications
tbl = Dataset[{<|"ID" -> 1, "CombinationPattern" -> "Parallel Race: Grammar + LLM", "Description" -> "Launch grammar-based parsing and one or more LLM interpreters in parallel; whichever yields a valid parse first is accepted.", "Pros" -> {"Fast when grammar succeeds", "Robust fallback", "Reduces latency unpredictability of LLMs"}, "ConsTradeoffs" -> {"Requires orchestration", "Need a validator for LLM output"}|>, <|"ID" -> 2, "CombinationPattern" -> "Grammar-First, LLM-Fallback", "Description" -> "Try grammar parser first; if it fails, invoke LLM-based parsing or normalization.", "Pros" -> {"Deterministic preference for grammar", "Testable correctness when grammar succeeds"}, "ConsTradeoffs" -> {"LLM fallback may produce inconsistent structures"}|>, <|"ID" -> 3, "CombinationPattern" -> "LLM-Assisted Grammar (Rule-Level)", "Description" -> "Individual grammar rules delegate to an LLM for ambiguous or context-heavy matching; LLM supplies tokens or AST fragments.", "Pros" -> {"Handles complexity without inflating grammar", "Modular LLM usage"}, "ConsTradeoffs" -> {"LLM behavior may break rule determinism", "Harder to reproduce"}|>, <|"ID" -> 4, "CombinationPattern" -> "LLM Normalizer -> Grammar Parser", "Description" -> "When grammar fails, LLM rewrites/normalizes input into a canonical form; grammar is applied again.", "Pros" -> {"Grammar remains simple", "Leverages LLM skill at paraphrasing"}, "ConsTradeoffs" -> {"Quality depends on normalizer", "Feedback loops possible"}|>, <|"ID" -> 5, "CombinationPattern" -> "Hybrid Declarative vs Procedural Parsing", "Description" -> "Grammar extracts structural/declarative parts; LLM interprets procedural/stepwise parts or vice versa.", "Pros" -> {"Each subsystem tackles what it's best at", "Reduces grammar complexity"}, "ConsTradeoffs" -> {"Harder to maintain global consistency", "Requires AST stitching logic"}|>, <|"ID" -> 6, "CombinationPattern" -> "Grammar-Generated Tests for LLM", "Description" -> "Grammar used to generate examples and counterexamples; LLM outputs are validated against grammar constraints.", "Pros" -> {"Powerful for verifying LLM outputs", "Reduces hallucinations"}, "ConsTradeoffs" -> {"Grammar must encode constraints richly", "Validation pipeline required"}|>, <|"ID" -> 7, "CombinationPattern" -> "LLM as Adaptive Heuristic for Grammar Ambiguities", "Description" -> "When grammar yields multiple parses, LLM chooses or ranks the \"most plausible\" AST.", "Pros" -> {"Improves disambiguation", "Good for underspecified workflows"}, "ConsTradeoffs" -> {"LLM can pick syntactically impossible interpretations"}|>, <|"ID" -> 8, "CombinationPattern" -> "LLM as Semantic Phase After Grammar", "Description" -> "Grammar creates an AST; LLM interprets semantics, fills in missing steps, or resolves vague ops.", "Pros" -> {"Clean separation of syntax vs semantics", "Grammar ensures correctness"}, "ConsTradeoffs" -> {"Semantic interpretation may drift from syntax"}|>, <|"ID" -> 9, "CombinationPattern" -> "Self-Healing Parse Loop", "Description" -> "Grammar fails -> LLM proposes corrections -> grammar retries -> if still failing, LLM creates full AST.","Pros" -> {"Iterative and robust", "Captures user intent progressively"}, "ConsTradeoffs" -> {"More expensive; risk of oscillation"}|>, <|"ID" -> 10, "CombinationPattern" -> "Grammar Embedding Inside Prompt Templates", "Description" -> "Grammar definitions serialized into the prompt, guiding the LLM to conform to the grammar (soft constraints).", "Pros" -> {"Faster than grammar execution in some cases", "Encourages consistent structure"}, "ConsTradeoffs" -> {"Weak guarantees", "LLM may ignore grammar"}|>, <|"ID" -> 11, "CombinationPattern" -> "LLM-Driven Grammar Induction or Refinement", "Description" -> "LLM suggests new grammar rules or transformations; developer approves; the grammar evolves over time.", "Pros" -> {"Faster grammar evolution", "Useful for new workflow languages"}, "ConsTradeoffs" -> {"Requires human QA", "Risk of regressing accuracy"}|>, <|"ID" -> 12, "CombinationPattern" -> "Regex Engine as LLM Guardrail", "Description" -> "Regex or token rules used to validate or filter LLM results before accepting them.", "Pros" -> {"Lightweight constraints", "Useful for quick prototyping"}, "ConsTradeoffs" -> {"Regex too weak for complex syntax"}|>}];
tbl = tbl[All, KeyDrop[#, "ID"] &];
tbl = tbl[All, ReplacePart[#, "Pros" -> ColumnForm[#Pros]] &];
tbl = tbl[All, ReplacePart[#, "ConsTradeoffs" -> ColumnForm[#ConsTradeoffs]] &];
tbl = tbl[All, Style[#, FontFamily -> "Times New Roman"] & /@ # &];
ResourceFunction["GridTableForm"][tbl]
#
CombinationPattern
Description
Pros
ConsTradeoffs
1
Parallel Race: Grammar + LLM
Launch grammar-based parsing and one or more LLM interpreters in parallel; whichever yields a valid parse first is accepted.
{{Fast when grammar succeeds}, {Robust fallback}, {Reduces latency unpredictability of LLMs}}
{{Requires orchestration}, {Need a validator for LLM output}}
2
Grammar-First, LLM-Fallback
Try grammar parser first; if it fails, invoke LLM-based parsing or normalization.
{{Deterministic preference for grammar}, {Testable correctness when grammar succeeds}}
{{LLM fallback may produce inconsistent structures}}
3
LLM-Assisted Grammar (Rule-Level)
Individual grammar rules delegate to an LLM for ambiguous or context-heavy matching; LLM supplies tokens or AST fragments.
{{Handles complexity without inflating grammar}, {Modular LLM usage}}
{{LLM behavior may break rule determinism}, {Harder to reproduce}}
4
LLM Normalizer -> Grammar Parser
When grammar fails, LLM rewrites/normalizes input into a canonical form; grammar is applied again.
{{Grammar remains simple}, {Leverages LLM skill at paraphrasing}}
{{Quality depends on normalizer}, {Feedback loops possible}}
5
Hybrid Declarative vs Procedural Parsing
Grammar extracts structural/declarative parts; LLM interprets procedural/stepwise parts or vice versa.
{{Each subsystem tackles what it’s best at}, {Reduces grammar complexity}}
{{Harder to maintain global consistency}, {Requires AST stitching logic}}
6
Grammar-Generated Tests for LLM
Grammar used to generate examples and counterexamples; LLM outputs are validated against grammar constraints.
{{Powerful for verifying LLM outputs}, {Reduces hallucinations}}
{{Grammar must encode constraints richly}, {Validation pipeline required}}
7
LLM as Adaptive Heuristic for Grammar Ambiguities
When grammar yields multiple parses, LLM chooses or ranks the “most plausible” AST.
{{Improves disambiguation}, {Good for underspecified workflows}}
{{LLM can pick syntactically impossible interpretations}}
8
LLM as Semantic Phase After Grammar
Grammar creates an AST; LLM interprets semantics, fills in missing steps, or resolves vague ops.
{{Clean separation of syntax vs semantics}, {Grammar ensures correctness}}
{{Semantic interpretation may drift from syntax}}
9
Self-Healing Parse Loop
Grammar fails -> LLM proposes corrections -> grammar retries -> if still failing, LLM creates full AST.
{{Iterative and robust}, {Captures user intent progressively}}
{{More expensive; risk of oscillation}}
10
Grammar Embedding Inside Prompt Templates
Grammar definitions serialized into the prompt, guiding the LLM to conform to the grammar (soft constraints).
{{Faster than grammar execution in some cases}, {Encourages consistent structure}}
{{Weak guarantees}, {LLM may ignore grammar}}
11
LLM-Driven Grammar Induction or Refinement
LLM suggests new grammar rules or transformations; developer approves; the grammar evolves over time.
{{Faster grammar evolution}, {Useful for new workflow languages}}
{{Requires human QA}, {Risk of regressing accuracy}}
12
Regex Engine as LLM Guardrail
Regex or token rules used to validate or filter LLM results before accepting them.
{{Lightweight constraints}, {Useful for quick prototyping}}
{{Regex too weak for complex syntax}}
Diversity reasons
The diversity of combinations in the table above arises because Raku grammars and LLMs occupy fundamentally different but highly complementary positions in the parsing spectrum.
Raku grammars provide determinism, speed, verifiability, and structural guarantees, but they require upfront design and struggle with ambiguity, informal input, and evolving specifications.
LLMs, in contrast, excel at normalization, semantic interpretation, ambiguity resolution, and adapting to fluid or poorly defined languages, yet they lack determinism, may hallucinate, and are slower.
When these two technologies meet, every architectural choice —ย who handles syntax, who handles semantics, who runs first, who validates whom, who repairs or refinesย — defines a distinct strategy.
Hence, the design space naturally expands into many valid hybrid patterns rather than a single “best” pipeline.
That said, the fallback pattern implemented below can be considered the “best option” from certain development perspectives because it is simple, effective, and has fast execution times.
This section demonstrates the use of three different translation methods:
Grammar-based parser-interpreter of computational workflows
LLM-based translator using few-shot learning with relevant DSL examples
Natural Language Processing (NLP) interpreter using code templates and LLMs to fill-in the corresponding parameters
The translators are ordered according of their faithfulness, most faithful first. It can be said that at the same time, the translators are ordered according to their coverage — widest coverage is by the last.
Grammar-based
Here a recommender pipeline specified with natural language commands is translated into Raku code of the package “ML::SparseMatrixRecommender” using a sub of the paclet “DSLTranslation”:
spec = "create from dsData; apply LSI functions IDF, None, Cosine; recommend by profile for passengerSex:male, and passengerClass:1st; join across using dsData; echo the pipeline value";
The function DSLTranslation uses a web service by default but if Raku and the package “DSL::Translators” are installed it can use the provided Command Line Interface (CLI):
LLM translations can be done using a set of from-to rules. This is the so-called few shot learning of LLMs. The paclet “DSLExamples” has a collection of such examples for different computational workflows. (Mostly ML at this point.) The examples are hierarchically organized by programming language and workflow name; see the resource file “dsl-examples.json”, or execute DSLExamples[].
Here is a table that shows the known DSL translation examples in “DSL::Examples” :
Here is a recommender pipeline specified with natural language commands:
spec = "new recommender; create from @dsData; apply LSI functions IDF, None, Cosine; recommend by profile for passengerSex:male, and passengerClass:1st; join across with @dsData on \"id\"; echo the pipeline value; classify by profile passengerSex:female, and passengerClass:1st on the tag passengerSurvival; echo value";
commands = StringSplit[spec, ""];
Translate to WL code line-by-line:
res = LLMPipelineSegment[] /@ commands; res = Map[StringTrim@StringReplace[#, RegularExpression["Output\h*:"] -> ""] &, res];
res = StringRiffle[res, "==>"]
tmplJudge = StringTemplate["Choose the generated code that most fully adheres to the spec:\\n\\n\\n`spec`\\n\\n\\nfrom the following `lang` generation results:\\n\\n\\n1) DSL-grammar:\\n`dslGrammar`\\n\\n\\n2) LLM-examples:\\n`llmExamples`\\n\\n\\n3) NLP-template-engine:\\n`nlpTemplateEngine`\\n\\n\\nand copy it:"]
spec = " make a brand new recommender with the data @dsData; apply LSI functions IDF, None, Cosine; recommend by profile for passengerSex:male, and passengerClass:1st; join across with @dsData on \"id\"; echo the pipeline value; ";
Instead of having DSL-grammar- and LLM computations running in parallel, we can make an LLM-graph in which the LLM computations are invoked if the DSL-grammar parsing-and-interpretation fails. In this section we make such a graph.
Before making the graph let us also generalize it to work with other ML workflows, not just recommendations.
Let us make an LLM function with a similar functionality. I.e. an LLM-function that classifies a natural language computation specification into workflow labels used by “DSLExamples”. Here is such a function using the sub LLMClassify provided by “NLPTemplateEngine”:
Here the LLM graph is run over a spec that can be parsed by DSL-grammar (notice the very short computation time):
Here is the obtained result:
Here is a spec that cannot be parsed by the DSL-grammar interpreter — note that there is just a small language change in the first line:
spec = " create from @dsData; apply LSI functions IDF, None, Cosine; recommend by profile for passengerSex:male, and passengerClass:1st; join across with @dsData on \"id\"; echo the pipeline value; ";
Nevertheless, we obtain a correct result via LLM-examples:
Let us specify another workflow — for ML-classification with Wolfram Language — and run the graph:
spec = " use the dataset @dsData; split the data into training and testing parts with 0.8 ratio; make a nearest neighbors classifier; show classifier accuracy, precision, and recall; echo the pipeline value; ";
Using LLM graphs gives the ability to impose desired orchestration and collaboration between deterministic programs and LLMs.
By contrast, the “inversion of control” of LLM – tools is “capricious.”
LLM-graphs are both a generalization of LLM-tools, and a lower level infrastructural functionality than LLM-tools.
The LLM-graph for the parallel-race translation is very similar to the LLM-graph for comprehensive document summarization described in [AA4].
The expectation that DSL examples would provide both fast and faithful results is mostly confirmed in โ20 experiments.
Using the NLP template engine is also fast because LLMs are harnessed through QAS.
The DSL examples translation had to be completed with a workflow classifier.
Such classifiers are also part of the implementations of the other two approaches .
The grammar – based one uses a deterministic classifier, [AA1]
The NLP template engine uses an LLM classifier .
An interesting extension of the current work is to have a grammar-LLM combination in which when the grammar fails then the LLM “normalizes” the specs until the grammar can parse them.
Currently, LLMGraph does not support graphs with cycles, hence this approach “can wait” (or be implemented by other means .)
Multiple DSL examples can be efficiently derived by random sentence generation with different grammars.
Similar to the DSL commands classifier making approach taken in [AA1] .
LLMs can be also used to improve and extend the DSL grammars.
And it is interesting to consider automating that process, instead of doing it via human supervision.
Program a few versions of circle chords based visualization routines.
Called chord trail plots below.
Marvel at chord trail plots for larger moduli.
Make multiple collections of them.
Look into number of primitive roots distributions.
Consider making animations of the collections.
The animations should not be โchaoticโ — they should have some inherent visual flow in them.
Consider different ways of sorting chord trail plots.
Using number theoretic arguments.
Yeah, would be nice, but requires too much head scratching and LLM-ing.
Convert plots to images and sort them.
Some might say that that is a โbrute forceโ application.
Simple image sort does not work.
Latent Semantic Analysis (LSA) application.
After failing to sort the chord trail image collections by โsimpleโ means, the idea applying LSA came to mind.
LSA being, of course, a favorite technique that was applied to sorting images multiple times in the past, in different contexts, [AAn1, AAn3, AAn4, AAn5, AAv3].
The function ChordTrails can be generalized to take a (pre-computed) chords argument. Here is an example of chords plot that connects integers that are modular inverses of each other:
The Doomsday Clock is a symbolic timepiece maintained by the Bulletin of the Atomic Scientists (BAS) since 1947. It represents how close humanity is perceived to be to global catastrophe, primarily nuclear war but also including climate change and biological threats. The clockโs hands are set annually to reflect the current state of global security; midnight signifies theoretical doomsday.
We take text data from the past announcements, and extract the Doomsday Clock reading statements.
Evolution of Doomsday Clock times
We extract relevant Doomsday Clock timeline data from the corresponding Wikipedia page.
(Instead of using a page from BAS.)
We show how timeline data from that Wikipedia page can be processed with โstandardโ Wolfram Language (WL) functions and with LLMs.
The result plot shows the evolution of the minutes to midnight.
The plot could show trends, highlighting significant global events that influenced the clock setting.
Hence, we put in informative callouts and tooltips.
The data extraction and visualization in the notebook serve educational purposes or provide insights into historical trends of global threats as perceived by experts. We try to make the ingestion and processing code universal and robust, suitable for multiple evaluations now or in the (near) future.
Remark: Keep in mind that the Doomsday Clock is a metaphor and its settings are not just data points but reflections of complex global dynamics (by certain experts and a board of sponsors.)
By observing the (plain) text of that page we see the Doomsday Clock time setting can be extracted from the sentence(s) that begin with the following phrase:
startPhrase = "Bulletin of the Atomic Scientists";
sentence = Select[Map[StringTrim, StringSplit[txtEN, "\n"]], StringStartsQ[#, startPhrase] &] // First
(*"Bulletin of the Atomic Scientists, with a clock reading 90 seconds to midnight"*)
Remark: The EBNF grammar above can be obtained with LLMs using a suitable prompt with example sentences. (We do not discuss that approach further in this notebook.)
Here the parsing functions are generated from the EBNF string above:
ClearAll["p*"]
res = GenerateParsersFromEBNF[ParseToEBNFTokens[ebnf]];
res // LeafCount
(*375*)
We must redefine the parser pANY (corresponding to the EBNF rule โโ) in order to prevent pANY of gobbling the word โclockโ and in that way making the parser pOPENING fail.
54jfnd 9y2f clock is reading 46 second to midnight
clock is reading 900 minutes to midnight
clock is reading 955 second to midnight
clock reading 224 minute to midnight
clock reading 410 minute to midnight
jdsf5at clock reading 488 seconds to midnight
Here is a verification table with correct- and incorrect spellings:
lsPhrases = {
"doomsday clock is reading 2 seconds to midnight",
"dooms day cloc is readding 2 minute and 22 sekonds to mildnight"};
ParsingTestTable[pCLOCKREADING, lsPhrases, "Layout" -> "Vertical"]
Parsing of numeric word forms
One way to make the parsing more robust is to implement the ability to parse integer names (or numeric word forms) not just integers.
There are two parsing results for โfifty sevenโ, because pWordedInteger is defined with p10sโpUpTo10โp10s… . This can be remedied by using ParseJust or ParseShortest:
Let us try the new parser using integer names for the clock time:
str = "the doomsday clock is reading two minutes and forty five seconds to midnight";
pTOP[ToTokens@str]
(*{{{}, {"Minutes" -> 2, "Seconds" -> 45}}}*)
Enhance with LLM parsing
There are multiple ways to employ LLMs for extracting โclock readingsโ from arbitrary statements for Doomsday Clock readings, readouts, and measures. Here we use LLM few-shot training:
flop = LLMExampleFunction[{
"the doomsday clock is reading two minutes and forty five seconds to midnight" -> "{\"Minutes\":2, \"Seconds\": 45}",
"the clock of the doomsday gives 92 seconds to midnight" -> "{\"Minutes\":0, \"Seconds\": 92}",
"The bulletin atomic scienist maybe is set to a minute an 3 seconds." -> "{\"Minutes\":1, \"Seconds\": 3}"
}, "JSON"]
Here is an example invocation:
flop["Maybe the doomsday watch is at 23:58:03"]
(*{"Minutes" -> 1, "Seconds" -> 57}*)
The following function combines the parsing with the grammar and the LLM example function — the latter is used for fallback parsing:
Here is the application of the combine function above over a certain โrandomโ Doomsday Clock statement:
s = "You know, sort of, that dooms-day watch is 1 and half minute be... before the big boom. (Of doom...)";
GetClockReading[s]
(*<|"Minutes" -> 1, "Seconds" -> 30|>*)
Remark: The same type of robust grammar-and-LLM combination is explained in more detail in the video “Robust LLM pipelines (Mathematica, Python, Raku)”, [AAv1]. (See, also, the corresponding notebook [AAn1].)
Timeline
In this section we extract Doomsday Clock timeline data and make a corresponding plot.
Here we get the Doomsday Clock timeline table from that page in JSON format using an LLM:
res =
LLMSynthesize[{
"Give the time table of the doomsday clock as a time series that is a JSON array.",
"Each element of the array is a dictionary with keys 'Year', 'MinutesToMidnight', 'Time', 'Description'.",
txtWk,
LLMPrompt["NothingElse"]["JSON"]
},
LLMEvaluator -> LLMConfiguration[<|"Provider" -> "OpenAI", "Model" -> "gpt-4o", "Temperature" -> 0.4, "MaxTokens" -> 5096|>]
]
(*"```json[{\"Year\": 1947, \"MinutesToMidnight\": 7, \"Time\": \"23:53\", \"Description\": \"The initial setting of the Doomsday Clock.\"},{\"Year\": 1949, \"MinutesToMidnight\": 3, \"Time\": \"23:57\", \"Description\": \"The Soviet Union tests its first atomic bomb, officially starting the nuclear arms race.\"}, ... *)
Post process the LLM result:
res2 = ToString[res, CharacterEncoding -> "UTF-8"];
res3 = StringReplace[res2, {"```json", "```"} -> ""];
res4 = ImportString[res3, "JSON"];
res4[[1 ;; 3]]
(*{{"Year" -> 1947, "MinutesToMidnight" -> 7, "Time" -> "23:53", "Description" -> "The initial setting of the Doomsday Clock."}, {"Year" -> 1949, "MinutesToMidnight" -> 3, "Time" -> "23:57", "Description" -> "The Soviet Union tests its first atomic bomb, officially starting the nuclear arms race."}, {"Year" -> 1953, "MinutesToMidnight" -> 2, "Time" -> "23:58", "Description" -> "The United States and the Soviet Union test thermonuclear devices, marking the closest approach to midnight until 2020."}}*)
Make a dataset with the additional column โDateโ (having date-objects):
Remark: The LLM derived descriptions above are shorter than the descriptions in the column โReasonโ of the dataset obtained parsing the page data. For the plot tooltips below we use the latter.
Timeline plot
In order to have informative Doomsday Clock evolution plot we obtain and partition datasetโs time series into step-function pairs:
Remark: By hovering with the mouse over the black points the corresponding descriptions can be seen. We considered using clock-gauges as tooltips, but showing clock-settings reasons is more informative.
As expected, parsing, plotting, or otherwise processing the Doomsday Clock settings and statements are excellent didactic subjects for textual analysis (or parsing) and temporal data visualization. The visualization could serve educational purposes or provide insights into historical trends of global threats as perceived by experts. (Remember, the clockโs settings are not just data points but reflections of complex global dynamics.)
One possible application of the code in this notebook is to make a โweb serviceโ that gives clock images with Doomsday Clock readings. For example, click on this button:
… or “Making Robust LLM Computational Pipelines from Software Engineering Perspective”
Abstract
Large Language Models (LLMs) are powerful tools with diverse capabilities, but from Software Engineering (SE) Point Of View (POV) they are unpredictable and slow. In this presentation we consider five ways to make more robust SE pipelines that include LLMs. We also consider a general methodological workflow for utilizing LLMs in “every day practice.”
Here are the five approaches we consider:
DSL for configuration-execution-conversion
Infrastructural, language-design level solution
Detailed, well crafted prompts
AKA “Prompt engineering”
Few-shot training with examples
Via a Question Answering System (QAS) and code templates
Grammar-LLM chain of responsibility
Testings with data types and shapes over multiple LLM results
Compared to constructing SE pipelines, Literate Programming (LP) offers a dual or alternative way to use LLMs. For that it needs support and facilitation of:
Convenient LLM interaction (or chatting)
Document execution (weaving and tangling)
The discussed LLM workflows methodology is supported in Python, Raku, Wolfram Language (WL). The support in R is done via Python (with “reticulate”, [TKp1].)
The presentation includes multiple examples and showcases.
Modeling of the LLM utilization process is hinted but not discussed.
All systematic approaches of unfolding and refining workflows based on LLM functions, will include several decision points and iterations to ensure satisfactory results.
This flowchart outlines such a systematic approach:
We do not examine the data source and we do not want to reason too much about the data using the stats. We started this notebook by just wanting to make the bubble charts (both 2D and 3D.) Nevertheless, we are tempted to say and justify statements like:
Here is the Pareto principle plot of for the number of created (or renamed) programming languages per creator (using the WFR function ParetoPrinciplePlot):
We can see that โ25% of the creators correspond to โ50% of the languages.
Popularity
Obviously, programmers can and do use more than one programming language. Nevertheless, it is interesting to see the Pareto principle plot for the languages “mind share” based on the number of users estimates.
Remark: Note that we “cheat” by adding 1 before taking the logarithms.
We obtain the tables of correlations plots using the newly introduced, experimental PairwiseListPlot. If we remove the rows with zeroes some of the correlations become more obvious. Here is the corresponding tab view of the two correlation tables:
Remark: In the 3D bubble chart plot “Mathematica” and “Wolfram Language” are easier to discern.
Second system effect traces
In this section we try — and fail — to demonstrate that the more programming languages a team of creators makes the less successful those languages are. (Maybe, because they are more cumbersome and suffer the Second system effect?)
Remark: This section is mostly made “for fun.” It is not true that each sets of languages per creators team is made of comparable languages. For example, complementary languages can be in the same set. (See, HTTP, HTML, URL.) Some sets are just made of the same language but with different names. (See, Perl 6 and Raku, and Mathematica and Wolfram Language.) Also, older languages would have the First mover advantage.
fWLForm = LLMSynthesize[{"Convert the following Markdown formatted text into a Mathematica formatted text using TextCell:", #, LLMPrompt["NothingElse"]["Mathematica"]}, LLMEvaluator -> LLMConfiguration["Model" -> "gpt-4"]] &;
Mathematica for prediction algorithms 
