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Go deh!Mainly Tech projects on Python and Electronic Design Automation.From all truths to (ir)relevancieshttp://paddy3118.blogspot.com/feeds/7018584544614148995/comments/default <div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI6Tg5CMmhqxo3YLYzw2sarTsTkBTDxuJ8nyTzdwG-C-yFwSB2Z-nu8EJcmRQM0Lw-XcY3obH4uBXrbaf9NXblIdLdcIM4lUfhDU8llOBIzzBdOJuh9TTJOhIaJXDiibJ9Tt8ZNV4ag9ee8Vb7lSs3oqy4w-5R4GAEgJvi3MThZYt-BXgUnWgt/s1024/Gemini_Generated_Image_ahywcpahywcpahyw.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1024" data-original-width="1024" height="508" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI6Tg5CMmhqxo3YLYzw2sarTsTkBTDxuJ8nyTzdwG-C-yFwSB2Z-nu8EJcmRQM0Lw-XcY3obH4uBXrbaf9NXblIdLdcIM4lUfhDU8llOBIzzBdOJuh9TTJOhIaJXDiibJ9Tt8ZNV4ag9ee8Vb7lSs3oqy4w-5R4GAEgJvi3MThZYt-BXgUnWgt/w508-h508/Gemini_Generated_Image_ahywcpahywcpahyw.png" width="508" /></a></div><div _ngcontent-ng-c2711534780="" class="markdown markdown-main-panel enable-updated-hr-color" dir="ltr" id="model-response-message-contentr_fbb1854db7be6292" style="--animation-duration: 400ms; --fade-animation-function: linear; font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px;"><div class="attachment-container generated-images" style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px;"><div _ngcontent-ng-c224903204="" class="loader" style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px;"> </div></div>Following up on <a href="https://paddy3118.blogspot.com/2025/07/all-truth-in-truthtables.html" rel="nofollow" target="_blank">my previous post</a> about truth tables, I now ask a subtler question: which inputs actually matter? Some variables, though present, leave no trace on the output. In this post, I uncover those quiet bits — the irrelevant inputs — and learn how to spot them with precision.<h4 style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Building on the previous</h4>The previous post showed that for a given number of inputs, there is a finite, but rapidly growing, number of possible truth tables. For two inputs, there are 16. For three inputs, there are 256. This leads to a powerful idea: we can create a standardized format for all truth tables and uniquely identify the parts in each one.<h3 style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px;">The Standardized Truth Table (STT) Format</h3><h3 style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px;"> </h3><h3 style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px;"><span style="-webkit-text-stroke-width: 0px; color: #800180; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0]<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">:<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #274e13; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">r<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">=================<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">:<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #04ff00; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">0<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">:<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #04ff00; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">0<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">1 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">:<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #04ff00; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">0<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">:<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #04ff00; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">1<span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">  </h3>Colour Key<ul style="text-align: left;"><li style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Inputs</li><li style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Input count</li><li style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Result, r </li><li style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Result vector: 0001 </li></ul>We have a standardized truth table with i inputs. Each row in the inputs section of the truth table is an increasing binary count from 0 to 2**i−1. The result column for the truth table has 2**i rows, leading to 2**(2**i) different possible truth table result vectors for i inputs.Any one possible result, r, of the standardized truth table is read as the binary bits of the output going from the topmost row (where the inputs are all zero) down to the bottom-most row, in order. This single binary number, the "result number" or vector, uniquely identifies the truth table and the boolean function it represents.<h3 style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px;">Irrelevant Variables in Boolean Expressions</h3>A variable in a boolean expression is considered irrelevant if its value has no effect on the final output. In a truth table, this means that changing the value of that input variable while all other inputs remain the same does not change the output.While this is easy to understand, it can be difficult to spot for complex functions. This is where the standardized truth table format, STT, comes in handy.<h3 style="font-family: Google Sans, sans-serif; line-height: 1.15; margin-top: 0px;">Irrelevancy Calculation</h3>For an i-input truth table with a 2**2**i bit result, r, we can efficiently check for irrelevant variables. The key insight is that for any single input variable, say i[n], the other input bits change in the exact same order when i[n]=0 as they do when i[n]=1, in the input count region of the STT.Therefore, if the result bits for the rows where i[n]=0 are the same as the result bits for the rows where i[n]=1, then the input variable i[n] has no effect on the output and is therefore irrelevant.Let's illustrate with an example for a 3-input truth table.Example <table><thead><tr><th>i[2]</th> <th>i[1]</th> <th>i[0]</th> <th>:</th> <th align="left">Output</th> </tr> </thead> <tbody> <tr> <td>0</td> <td>0</td> <td>0</td> <td>:</td> <td align="left">r0</td> </tr> <tr> <td>0</td> <td>0</td> <td>1</td> <td>:</td> <td align="left">r1</td> </tr> <tr> <td>0</td> <td>1</td> <td>0</td> <td>:</td> <td align="left">r2</td> </tr> <tr> <td>0</td> <td>1</td> <td>1</td> <td>:</td> <td align="left">r3</td> </tr> <tr> <td>1</td> <td>0</td> <td>0</td> <td>:</td> <td align="left">r4</td> </tr> <tr> <td>1</td> <td>0</td> <td>1</td> <td>:</td> <td align="left">r5</td> </tr> <tr> <td>1</td> <td>1</td> <td>0</td> <td>:</td> <td align="left">r6</td> </tr> <tr> <td>1</td> <td>1</td> <td>1</td> <td>:</td> <td align="left">r7</td></tr></tbody></table> Consider checking if i[1] is irrelevant. The values of the other inputs (i[2] and i[0]) follow the sequence 00, 01, 10, 11 both when i[1] is 0 and when i[1] is 1.<ul style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px; padding-inline-start: 32px;"><li style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px;">When i[1]=0, the corresponding output bits are r0,r1,r4,r5.</li><li style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px;">When i[1]=1, the corresponding output bits are r2,r3,r6,r7.</li></ul>If the sequence of bits (r0,r1,r4,r5) is identical to the sequence of bits (r2,r3,r6,r7), then the input variable i[1] has no effect on the output and is therefore irrelevant.This method allows for a very efficient, algorithmic approach to simplifying boolean expressions. <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>from collections import defaultdict</div><div>import pprint</div> <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"> <div>def is_irrelevant(considered: int,</div><div>                  input_count:int,</div><div>                  result_vector: int) -> bool:</div><div>    """</div><div>    Determine whether a specific input variable is irrelevant to the output of the truth table.</div><div>    </div><div>    Args:</div><div>        considered: Index of the input variable to test.</div><div>        input_count: Total number of input variables.</div><div>        result_vector: Integer representing the output bits of the truth table.</div><div>    </div><div>    Returns:</div><div>        True if the input is irrelevant, False otherwise.</div><div>    """</div><div>    considered_to_resultbits = {0: [], 1: []}</div> <div>    for in_count in range(2**input_count):</div><div>        considered_bit = (in_count >> considered) & 1</div><div>        resultbit = (result_vector >> in_count) & 1</div><div>        considered_to_resultbits[considered_bit].append(resultbit)</div><div>    </div><div>    return considered_to_resultbits[0] == considered_to_resultbits[1]</div></div> <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"> <div>def find_irrelevant_ttable_inputs(input_count:int,</div><div>                                  result_vector: int) -> list[int]:</div><div>    """</div><div>    Identify which input variables are irrelevant in a standardized truth table.</div><div>    </div><div>    Args:</div><div>        input_count: Number of input variables.</div><div>        result_vector: Integer representing the output bits of the truth table.</div><div>    </div><div>    Returns:</div><div>        List of input indices that are irrelevant.</div><div>    """</div><div>    irrelevant = [i for i in range(input_count)</div><div>                  if is_irrelevant(i, input_count, result_vector)]</div><div>    </div><div>    return irrelevant</div> </div></div> <h3 style="font-family: Google Sans Text, sans-serif; line-height: 1.15; margin-top: 0px; text-align: left;">Relevant and irrelevant result vectors for STT's</h3>I am interested in the truthtables/boolean expressions of an increasing number of inputs. Previously I was taking all the zero input STT, then all the 1-input, all the 2-input, ...  That had repetitions and irrelevancies. I can now take just the relevant result vectors for each case, or, take the maximum inputs I can handle and sort the result vectors so that those with the most irrelevancies for that maximum number of inputs, come first.Here's the code I used to investigate these properties of irrelevances: <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>def print_STT_results_sensitive_to_inputs(max_input_count: int) -> None:</div><div>    """</div><div>    Print a summary of irrelevance patterns across all standardized truth tables (STTs)</div><div>    for input counts from 0 up to `max_input_count - 1`.</div> <div>    For each input count `i`, the function:</div><div>      - Computes all possible result vectors (2**(2**i))</div><div>      - Identifies which inputs are irrelevant for each result</div><div>      - Summarizes how many result vectors have at least one irrelevant input</div><div>      - Prints spacing patterns (diffs) between result indices with irrelevance</div><div>      - Checks whether irrelevance indices for input count `i` begin with those from `i-1`</div> <div>    Args:</div><div>        max_input_count: The exclusive upper bound on input counts to analyze.</div><div>    """</div> <div>    imax2irrelevances = {}</div><div>    for imax in range(max_input_count):</div><div>        print(f"\nINPUTS = {imax}\n==========")</div><div>        result_count = 2**2**imax</div><div>        print(f"  total_possible_tt_results =", result_count)</div> <div>        result2irrelevances = {r:[i for i in range(imax)</div><div>                                  if is_irrelevant(i, imax, r)]</div><div>                               for r in range(result_count)}</div><div>        txt = pprint.pformat(result2irrelevances, compact=True, indent=2).replace('\n', '\n  ')</div><div>        txt = txt.replace('{', '{\n   ')</div><div>        #print("  result2irrelevances = ", txt)</div> <div>        irrelevances = [k for k, v in result2irrelevances.items()</div><div>                        if v]</div><div>        </div><div>        imax2irrelevances[imax] = irrelevances</div> <div>        relevances = result_count - len(irrelevances)</div><div>        print(f"  {irrelevances = }")</div><div>        print(f"    {len(irrelevances) = }, {relevances = }")</div> <div>        table = chop_line(format_irrelevance_table_full(result2irrelevances, imax)[0], 220)</div><div>        print("\nSTT Result Irrelevance vs Input Table"</div><div>              f"\n{table}")</div> <div>        # First-order differences between irrelevance indices</div><div>        diff0 = [irrelevances[j] - irrelevances[j-1] </div><div>                 for j in range(1, len(irrelevances))]</div><div>        #print(f"    {diff0 = }")</div> <div>        # Second-order differences between irrelevance indices</div><div>        diff1 = [diff0[j] - diff0[j-1] </div><div>                 for j in range(1, len(diff0))]</div><div>        #print(f"    {diff1 = }")</div><div>    </div><div>    print("\n\nIrrelevance indices reflected about the center of the r count?"</div><div>          f"\n  {is_irrelevance_reflected(imax2irrelevances)}")  # True so far</div> <div>    print('Irrelevances for `i` inputs begin with those from `i-1`?')</div><div>    previous_prefixed = all((i0 := imax2irrelevances[i-1]) == imax2irrelevances[i][:len(i0)]</div><div>                            for i in range(1, max_input_count))</div><div>    print(f"  {previous_prefixed}")  # True so far</div> <div>def is_irrelevance_reflected(ins2irrel: dict[int, list[int]]) -> bool:</div><div>    """</div><div>    Check whether irrelevance indices are symmetric about the center of the result vector space.</div> <div>    For each input count `i`, the result vector space has size 2**(2**i).</div><div>    This function checks whether for every irrelevance index `r < half_range`,</div><div>    its mirror index `full_range - 1 - r` is also marked as irrelevant.</div> <div>    Args:</div><div>        ins2irrel: Mapping from input count to list of result indices with irrelevance.</div> <div>    Returns:</div><div>        True if all irrelevance sets are symmetric about the center, False otherwise.</div><div>    """</div><div>    for imax, irrelevances in ins2irrel.items():</div><div>        irrel_set = set(irrelevances)</div><div>        full_range = 2**2**imax</div><div>        half_range = full_range / 2</div> <div>        if not all((full_range - 1 - i) in irrel_set for i in irrel_set if i < half_range):</div><div>            return False</div><div>    return True</div> <div>def format_irrelevance_table(result2irrelevances: dict[int, list[int]], input_count: int) -> tuple[str, str]:</div><div>    """</div><div>    Create a compact table showing which inputs are irrelevant for each result vector `r`.</div> <div>    Each column is sized to match the width of its corresponding `r` value.</div><div>    Only includes columns for `r` values that have at least one irrelevant input.</div><div>    Each row corresponds to an input index, with '@' if that input is irrelevant for that `r`, else blank.</div> <div>    Returns:</div><div>        A tuple of (plain text table string, markdown table string)</div><div>    """</div><div>    if input_count == 0:</div><div>        msg = "No inputs to analyze (input_count = 0)."</div><div>        md = "| Input | (none) |\n|-------|--------|\n| (none) |        |"</div><div>        return msg, md</div> <div>    filtered_r = [r for r, irrels in result2irrelevances.items() if irrels]</div><div>    if not filtered_r:</div><div>        msg = "No irrelevant inputs found for any result vector."</div><div>        md = "| Input | (none) |\n|-------|--------|\n" + "\n".join(</div><div>            f"| i[{i}] |        |" for i in range(input_count)</div><div>        )</div><div>        return msg, md</div> <div>    # Determine individual column widths based on r string length</div><div>    r_labels = [str(r) for r in filtered_r]</div><div>    col_widths = [len(label) for label in r_labels]</div> <div>    # Header row</div><div>    header_plain = " " * 8 + " ".join(label.rjust(w) for label, w in zip(r_labels, col_widths))</div><div>    header_md = "| Input | " + " | ".join(label.rjust(w) for label, w in zip(r_labels, col_widths)) + " |"</div> <div>    # Markdown separator row</div><div>    separator_md = "|-------|" + "|".join("-" * (w + 2) for w in col_widths) + "|"</div> <div>    # Rows</div><div>    rows_plain = []</div><div>    rows_md = []</div><div>    for i in range(input_count):</div><div>        label = f"i[{i}]".ljust(8)</div><div>        row_plain = label + " ".join("@" .rjust(w) if i in result2irrelevances[r] else " " * w</div><div>                                     for r, w in zip(filtered_r, col_widths))</div><div>        row_md = "| " + f"i[{i}]".ljust(5) + " | " + " | ".join("@" .rjust(w) if i in result2irrelevances[r] else " " * w</div><div>                                                               for r, w in zip(filtered_r, col_widths)) + " |"</div><div>        rows_plain.append(row_plain)</div><div>        rows_md.append(row_md)</div> <div>    plain_table = "\n".join([header_plain] + rows_plain)</div><div>    markdown_table = "\n".join([header_md, separator_md] + rows_md)</div> <div>    return plain_table, markdown_table</div> <div>def format_irrelevance_table_full(result2irrelevances: dict[int, list[int]], input_count: int) -> tuple[str, str]:</div><div>    """</div><div>    Create a full table showing which inputs are irrelevant for each result vector `r`.</div> <div>    Includes columns for all result vector indices (0 to max(r)).</div><div>    Each row corresponds to an input index, with '@' if that input is irrelevant for that `r`, else blank.</div> <div>    Returns:</div><div>        A tuple of (plain text table string, markdown table string)</div><div>    """</div><div>    if input_count == 0:</div><div>        msg = "No inputs to analyze (input_count = 0)."</div><div>        md = "| Input | (none) |\n|-------|--------|\n| (none) |        |"</div><div>        return msg, md</div> <div>    all_r = sorted(result2irrelevances.keys())</div><div>    r_labels = [str(r) for r in all_r]</div><div>    col_widths = [len(label) for label in r_labels]</div> <div>    # Header row</div><div>    header_plain = " " * 8 + " ".join(label.rjust(w) for label, w in zip(r_labels, col_widths))</div><div>    header_md = "| Input | " + " | ".join(label.rjust(w) for label, w in zip(r_labels, col_widths)) + " |"</div> <div>    # Markdown separator row</div><div>    separator_md = "|-------|" + "|".join("-" * (w + 2) for w in col_widths) + "|"</div> <div>    # Rows</div><div>    rows_plain = []</div><div>    rows_md = []</div><div>    for i in range(input_count):</div><div>        label = f"i[{i}]".ljust(8)</div><div>        row_plain = label + " ".join("@" .rjust(w) if i in result2irrelevances.get(r, []) else " " * w</div><div>                                     for r, w in zip(all_r, col_widths))</div><div>        row_md = "| " + f"i[{i}]".ljust(5) + " | " + " | ".join("@" .rjust(w) if i in result2irrelevances.get(r, []) else " " * w</div><div>                                                               for r, w in zip(all_r, col_widths)) + " |"</div><div>        rows_plain.append(row_plain)</div><div>        rows_md.append(row_md)</div> <div>    plain_table = "\n".join([header_plain] + rows_plain)</div><div>    markdown_table = "\n".join([header_md, separator_md] + rows_md)</div> <div>    return plain_table, markdown_table</div> <div>def chop_line(table_text: str, max_length: int) -> str:</div><div>    """</div><div>    Chop each line in a multi-line string to a maximum length.</div><div>    If a line exceeds `max_length`, it is truncated to `max_length - 3` and '...' is appended.</div> <div>    Args:</div><div>        table_text: The full multi-line string to process.</div><div>        max_length: The maximum allowed line length.</div> <div>    Returns:</div><div>        A new multi-line string with long lines chopped and marked with '...'.</div><div>    """</div><div>    chopped_lines = []</div><div>    for line in table_text.splitlines():</div><div>        if len(line) > max_length:</div><div>            chopped_lines.append(line[:max_length - 3] + '...')</div><div>        else:</div><div>            chopped_lines.append(line)</div><div>    return '\n'.join(chopped_lines)</div> <div>print_STT_results_sensitive_to_inputs(5)</div><div> </div></div> </div><h3 style="text-align: left;">Irrelevances: Output</h3> <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>INPUTS = 0</div><div>==========</div><div>  total_possible_tt_results = 2</div><div>  irrelevances = []</div><div>    len(irrelevances) = 0, relevances = 2</div> <div>STT Result Irrelevance vs Input Table</div><div>No inputs to analyze (input_count = 0).</div> <div>INPUTS = 1</div><div>==========</div><div>  total_possible_tt_results = 4</div><div>  irrelevances = [0, 3]</div><div>    len(irrelevances) = 2, relevances = 2</div> <div>STT Result Irrelevance vs Input Table</div><div>        0 1 2 3</div><div>i[0]    @     @</div> <div>INPUTS = 2</div><div>==========</div><div>  total_possible_tt_results = 16</div><div>  irrelevances = [0, 3, 5, 10, 12, 15]</div><div>    len(irrelevances) = 6, relevances = 10</div> <div>STT Result Irrelevance vs Input Table</div><div>        0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15</div><div>i[0]    @     @                    @        @</div><div>i[1]    @         @          @              @</div> <div>INPUTS = 3</div><div>==========</div><div>  total_possible_tt_results = 256</div><div>  irrelevances = [0, 3, 5, 10, 12, 15, 17, 34, 48, 51, 60, 63, 68, 80, 85, 90, 95, 102, 119, 136, 153, 160, 165, 170, 175, 187, 192, 195, 204, 207, 221, 238, 240, 243, 245, 250, 252, 255]</div><div>    len(irrelevances) = 38, relevances = 218</div> <div>STT Result Irrelevance vs Input Table</div><div>        0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 ...</div><div>i[0]    @     @                    @        @                                                                                                  @        @                          @        @                            ...</div><div>i[1]    @         @          @              @                                                                                                                                                                            ...</div><div>i[2]    @                                         @                                                  @                                                  @                                                  @             ...</div> <div>INPUTS = 4</div><div>==========</div><div>  total_possible_tt_results = 65536</div><div>  irrelevances = [0, 3, 5, 10, 12, 15, 17, 34, 48, 51, 60, 63, 68, 80, 85, 90, 95, 102, 119, 136, 153, 160, 165, 170, 175, 187, 192, 195, 204, 207, 221, 238, 240, 243, 245, 250, 252, 255, 257, 514, 768, 771, 780, 783, 816, 819, 828, 831, 960, 963, 972, 975, 1008, 1011, 1020, 1023, 1028, 1280, 1285, 1290, 1295, 1360, 1365, 1370, 1375, 1440, 1445, 1450, 1455, 1520, 1525, 1530, 1535, 1542, 1799, 2056, 2313, 2560, 2565, 2570, 2575, 2640, 2645, 2650, 2655, 2720, 2725, 2730, 2735, 2800, 2805, 2810, 2815, 2827, 3072, 3075, 3084, 3087, 3120, 3123, 3132, 3135, 3264, 3267, 3276, 3279, 3312, 3315, 3324, 3327, 3341, 3598, 3840, 3843, 3845, 3850, 3852, 3855, 3888, 3891, 3900, 3903, 3920, 3925, 3930, 3935, 4000, 4005, 4010, 4015, 4032, 4035, 4044, 4047, 4080, 4083, 4085, 4090, 4092, 4095, 4112, 4352, 4369, 4386, 4403, 4420, 4437, 4454, 4471, 4488, 4505, 4522, 4539, 4556, 4573, 4590, 4607, 4626, 4883, 5140, 5397, 5654, 5911, 6168, 6425, 6682, 6939, 7196, 7453, 7710, 7967, 8224, 8481, 8704, 8721, 8738, 8755, 8772, 8789, 8806, 8823, 8840, 8857, 8874, 8891, 8908, 8925, 8942, 8959, 8995, 9252, 9509, 9766, 10023, 10280, 10537, 10794, 11051, 11308, 11565, 11822, 12079, 12288, 12291, 12300, 12303, 12336, 12339, 12348, 12351, 12480, 12483, 12492, 12495, 12528, 12531, 12540, 12543, 12593, 12850, 13056, 13059, 13068, 13071, 13073, 13090, 13104, 13107, 13116, 13119, 13124, 13141, 13158, 13175, 13192, 13209, 13226, 13243, 13248, 13251, 13260, 13263, 13277, 13294, 13296, 13299, 13308, 13311, 13364, 13621, 13878, 14135, 14392, 14649, 14906, 15163, 15360, 15363, 15372, 15375, 15408, 15411, 15420, 15423, 15552, 15555, 15564, 15567, 15600, 15603, 15612, 15615, 15677, 15934, 16128, 16131, 16140, 16143, 16176, 16179, 16188, 16191, 16320, 16323, 16332, 16335, 16368, 16371, 16380, 16383, 16448, 16705, 16962, 17219, 17408, 17425, 17442, 17459, 17476, 17493, 17510, 17527, 17544, 17561, 17578, 17595, 17612, 17629, 17646, 17663, 17733, 17990, 18247, 18504, 18761, 19018, 19275, 19532, 19789, 20046, 20303, 20480, 20485, 20490, 20495, 20560, 20565, 20570, 20575, 20640, 20645, 20650, 20655, 20720, 20725, 20730, 20735, 20817, 21074, 21331, 21588, 21760, 21765, 21770, 21775, 21777, 21794, 21811, 21828, 21840, 21845, 21850, 21855, 21862, 21879, 21896, 21913, 21920, 21925, 21930, 21935, 21947, 21964, 21981, 21998, 22000, 22005, 22010, 22015, 22102, 22359, 22616, 22873, 23040, 23045, 23050, 23055, 23120, 23125, 23130, 23135, 23200, 23205, 23210, 23215, 23280, 23285, 23290, 23295, 23387, 23644, 23901, 24158, 24320, 24325, 24330, 24335, 24400, 24405, 24410, 24415, 24480, 24485, 24490, 24495, 24560, 24565, 24570, 24575, 24672, 24929, 25186, 25443, 25700, 25957, 26112, 26129, 26146, 26163, 26180, 26197, 26214, 26231, 26248, 26265, 26282, 26299, 26316, 26333, 26350, 26367, 26471, 26728, 26985, 27242, 27499, 27756, 28013, 28270, 28527, 28784, 29041, 29298, 29555, 29812, 30069, 30326, 30464, 30481, 30498, 30515, 30532, 30549, 30566, 30583, 30600, 30617, 30634, 30651, 30668, 30685, 30702, 30719, 30840, 31097, 31354, 31611, 31868, 32125, 32382, 32639, 32896, 33153, 33410, 33667, 33924, 34181, 34438, 34695, 34816, 34833, 34850, 34867, 34884, 34901, 34918, 34935, 34952, 34969, 34986, 35003, 35020, 35037, 35054, 35071, 35209, 35466, 35723, 35980, 36237, 36494, 36751, 37008, 37265, 37522, 37779, 38036, 38293, 38550, 38807, 39064, 39168, 39185, 39202, 39219, 39236, 39253, 39270, 39287, 39304, 39321, 39338, 39355, 39372, 39389, 39406, 39423, 39578, 39835, 40092, 40349, 40606, 40863, 40960, 40965, 40970, 40975, 41040, 41045, 41050, 41055, 41120, 41125, 41130, 41135, 41200, 41205, 41210, 41215, 41377, 41634, 41891, 42148, 42240, 42245, 42250, 42255, 42320, 42325, 42330, 42335, 42400, 42405, 42410, 42415, 42480, 42485, 42490, 42495, 42662, 42919, 43176, 43433, 43520, 43525, 43530, 43535, 43537, 43554, 43571, 43588, 43600, 43605, 43610, 43615, 43622, 43639, 43656, 43673, 43680, 43685, 43690, 43695, 43707, 43724, 43741, 43758, 43760, 43765, 43770, 43775, 43947, 44204, 44461, 44718, 44800, 44805, 44810, 44815, 44880, 44885, 44890, 44895, 44960, 44965, 44970, 44975, 45040, 45045, 45050, 45055, 45232, 45489, 45746, 46003, 46260, 46517, 46774, 47031, 47288, 47545, 47802, 47872, 47889, 47906, 47923, 47940, 47957, 47974, 47991, 48008, 48025, 48042, 48059, 48076, 48093, 48110, 48127, 48316, 48573, 48830, 49087, 49152, 49155, 49164, 49167, 49200, 49203, 49212, 49215, 49344, 49347, 49356, 49359, 49392, 49395, 49404, 49407, 49601, 49858, 49920, 49923, 49932, 49935, 49968, 49971, 49980, 49983, 50112, 50115, 50124, 50127, 50160, 50163, 50172, 50175, 50372, 50629, 50886, 51143, 51400, 51657, 51914, 52171, 52224, 52227, 52236, 52239, 52241, 52258, 52272, 52275, 52284, 52287, 52292, 52309, 52326, 52343, 52360, 52377, 52394, 52411, 52416, 52419, 52428, 52431, 52445, 52462, 52464, 52467, 52476, 52479, 52685, 52942, 52992, 52995, 53004, 53007, 53040, 53043, 53052, 53055, 53184, 53187, 53196, 53199, 53232, 53235, 53244, 53247, 53456, 53713, 53970, 54227, 54484, 54741, 54998, 55255, 55512, 55769, 56026, 56283, 56540, 56576, 56593, 56610, 56627, 56644, 56661, 56678, 56695, 56712, 56729, 56746, 56763, 56780, 56797, 56814, 56831, 57054, 57311, 57568, 57825, 58082, 58339, 58596, 58853, 59110, 59367, 59624, 59881, 60138, 60395, 60652, 60909, 60928, 60945, 60962, 60979, 60996, 61013, 61030, 61047, 61064, 61081, 61098, 61115, 61132, 61149, 61166, 61183, 61423, 61440, 61443, 61445, 61450, 61452, 61455, 61488, 61491, 61500, 61503, 61520, 61525, 61530, 61535, 61600, 61605, 61610, 61615, 61632, 61635, 61644, 61647, 61680, 61683, 61685, 61690, 61692, 61695, 61937, 62194, 62208, 62211, 62220, 62223, 62256, 62259, 62268, 62271, 62400, 62403, 62412, 62415, 62448, 62451, 62460, 62463, 62708, 62720, 62725, 62730, 62735, 62800, 62805, 62810, 62815, 62880, 62885, 62890, 62895, 62960, 62965, 62970, 62975, 63222, 63479, 63736, 63993, 64000, 64005, 64010, 64015, 64080, 64085, 64090, 64095, 64160, 64165, 64170, 64175, 64240, 64245, 64250, 64255, 64507, 64512, 64515, 64524, 64527, 64560, 64563, 64572, 64575, 64704, 64707, 64716, 64719, 64752, 64755, 64764, 64767, 65021, 65278, 65280, 65283, 65285, 65290, 65292, 65295, 65297, 65314, 65328, 65331, 65340, 65343, 65348, 65360, 65365, 65370, 65375, 65382, 65399, 65416, 65433, 65440, 65445, 65450, 65455, 65467, 65472, 65475, 65484, 65487, 65501, 65518, 65520, 65523, 65525, 65530, 65532, 65535]</div><div>    len(irrelevances) = 942, relevances = 64594</div> <div>STT Result Irrelevance vs Input Table</div><div>        0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 ...</div><div>i[0]    @     @                    @        @                                                                                                  @        @                          @        @                            ...</div><div>i[1]    @         @          @              @                                                                                                                                                                            ...</div><div>i[2]    @                                         @                                                  @                                                  @                                                  @             ...</div><div>i[3]    @                                                                                                                                                                                                                ...</div> <div>Irrelevance indices reflected about the center of the r count?</div><div>  True</div><div>Irrelevances for `i` inputs begin with those from `i-1`?</div><div>  True</div> </div><h3 style="text-align: left;">OEIS</h3>The sequence of relevances: 2, 2, 10, 218, 64594 is already present as <a href="https://oeis.org/A000371" rel="nofollow" target="_blank">A000371 </a>on OEIS.The sequences of irelevances: 0, 2, 6, 38, 942 had no exact match although <a href="https://oeis.org/search?q=2%2c6%2c38%2c942%20id:A005530" target="_blank">A005530 </a>came close END. Sun, 14 Sep 2025 12:51:00 Go Deh celebrations!http://paddy3118.blogspot.com/feeds/3066718111574923076/comments/default <div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghCYUXmcR6-kfrZKbVof_gfZfqHnOE_cfxQx9l6Iii9pjsuFXpRuudMkEU5_gjPHNfsa34AFbNCefCkvGwfEOaCKGqXGzQfTZTsdoG9YUXkj9oGDfRoozESPySqOszV5PquQUtHDHG2-2snTbL3TAdo1FLEfeemLpuxJXHU7NFbUPKpagEmM48/s1024/A%20jubilant%20Jamaican%202.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1024" data-original-width="1024" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghCYUXmcR6-kfrZKbVof_gfZfqHnOE_cfxQx9l6Iii9pjsuFXpRuudMkEU5_gjPHNfsa34AFbNCefCkvGwfEOaCKGqXGzQfTZTsdoG9YUXkj9oGDfRoozESPySqOszV5PquQUtHDHG2-2snTbL3TAdo1FLEfeemLpuxJXHU7NFbUPKpagEmM48/s320/A%20jubilant%20Jamaican%202.png" width="320" /></a></div> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_z_l56oVliXpceL0jsn8lbb_wjVN9KF7c4vD0IAzKv76gWTvREQCqDvXJQa-Y7UlxdTCUbSYja7bSwPmRvf5yTo_E4sJW61CZTNdC2-utWn28tI0RuEPhRCbQqfb73_LddRGry9Sa7bNhAp6eDx8x4clBJAkm3__Tjr39byNDqiRlNdJFkQAi/s2048/Gemini_Generated_Image_pvdkv4pvdkv4pvdk.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2048" data-original-width="2048" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_z_l56oVliXpceL0jsn8lbb_wjVN9KF7c4vD0IAzKv76gWTvREQCqDvXJQa-Y7UlxdTCUbSYja7bSwPmRvf5yTo_E4sJW61CZTNdC2-utWn28tI0RuEPhRCbQqfb73_LddRGry9Sa7bNhAp6eDx8x4clBJAkm3__Tjr39byNDqiRlNdJFkQAi/s320/Gemini_Generated_Image_pvdkv4pvdkv4pvdk.png" width="320" /></a></div>  This year it's: <ol style="text-align: left;"><li>Fifty years since I first learned to program!</li><li>Thirty years of programming Python!</li><li>And In July, this blog topped one million views since its inception! </li></ol>I first learned to program by crashing an after-school class I saw the private school geeks disappearing to from Central library  in Nottingham. One of the Trent Polytechnic lecturers I now see sometimes on Computerphile - he mentioned how the council payed something towards the cost of the computer on the understanding that they have a class for students, but somehow only the guys from the fee paying Nottingham High School for Boys seemed to get the memo, hmm. That was s long ago, Paper tape and teletypes.Forward to 1995: A new job, amongst engineers where the few of them that could program, programmed in Perl. Perl was big within the chip design industry back then, but Aargh! I thought Perl was awful - for me it did not gel! I went looking for an alternative and found this little known language called Python.  The community on comp.lang.python seemed so nice, they valued readability and "One obvious way to do it" over Perls more than one way. I was hooked, and did my little bit of evangelizing. (OK, I was known as that Python guy - but they couldn't fault the code).Was it eleven years ago that I started the blog? It has been one of two main places where I have shown my work - the other is on site<a href="https://rosettacode.org/wiki/User:Paddy3118" rel="nofollow" target="_blank"> rosettacode.org</a> which, no doubt, has been scraped to death for AI's - it is one of the few places online showing different programming languages solving the same task. I wrote 199 of those RosettaCode tasks over the years, and many more of the examples solving those tasks; (mainly  Python and Awk language examples). Most, if not all of those task descriptions were my own work - not copied off other sites - if there are similarities in text then they probably copied from me or the task is so simple that there is going to be convergence). Lately, as you can tell from the pictures above, I have been dabbling in AI. AI is the future, but who knows how AI will affect that future? I am a programmer who can describe coding tasks to other people for them to complete the task. AI is an ever changing field, I think that I am learning to use it better, as it gets better too - I am both more and less precise in my prompting and finding out what the AI remembers and does best. I experiment with the  amount of description in the code to cut and paste into new sessions to cut the mistakes and continue from a clean base. I experiment with coding from pseudo-code. I have found that some abilities are universal: when exploring the use of SQL - which I am a hunt-n-peck coder in, I can still spot redundancies "You recompute the first query as part of the second and the second as part of the third, can't you reuse prior results", lead to the AI introducing common table expressions I think it was, and telling me all about them.I'm still scripting  😊 Sun, 10 Aug 2025 17:22:00 All Truth in Truthtables!http://paddy3118.blogspot.com/feeds/1386948987337358897/comments/default  <img alt="" height="236" src="https://blogger.googleusercontent.com/img/a/AVvXsEjAkKWov5Vkf0RC3lLoj7mdzZPEW2RRK5fqNDoD1wzaRXh8UUXI9pd3gYbneEhapx8xw2MprxeoN3Zmb1E-5scChoMYLZgcygDR8cwG7BaQtEZkWeYlMn8RUPWTWX4jerFY2kU0NR_AkSUfjVx85ifTtOIhzJq1ZQT_IXVIqtqJ3qBATtr2l0w4=w236-h236" width="236" />  (Best viewed on a larger than phone screen)To crib from my <a href="https://rosettacode.org/wiki/Truth_table" rel="nofollow" target="_blank">RosettaCode tasks</a> description and examples:<blockquote>A <a class="extiw" href="https://en.wikipedia.org/wiki/Truth_table" title="wp:Truth table">truth table</a> is a display of the inputs to, and the output of a Boolean equation organised as a table where each row gives one combination of input values and the corresponding value of the equation.</blockquote>And as examples:<pre><blockquote>Boolean expression: A ^ B A B : A ^ B 0 0 : 0 0 1 : 1 1 0 : 1 1 1 : 0 Boolean expression: S | ( T ^ U ) S T U : S | ( T ^ U ) 0 0 0 : 0 0 0 1 : 1 0 1 0 : 1 0 1 1 : 0 1 0 0 : 1 1 0 1 : 1 1 1 0 : 1 1 1 1 : 1</blockquote></pre><h2 style="text-align: left;"> Format</h2>A truth table has a header row of columns showing first the names of inputs assigned to each column; a visual separator - e.g. ':'; then the column name for the output result.The body of the table, under the inputs section, contains rows of all binary combinations of the inputs. It is usually arranged as each row of the input section being a binary count from zero to 2**input_count - 1  The body of the table, under the result section, contains rows showing the binary output produced from the input configuration in the same row, to the left.<h3 style="text-align: left;">Format used</h3> I am interested in the number of inputs rather than their names so will show vector i with the most significant indices to the left, (so the binary count in the input sections body looks right).Similarly I am interested in the bits in the result column rather than a name so will just call the result column r.<h2 style="text-align: left;">From one result to many</h2> Here's the invocation, and truth tables produced for some simple boolean operators:<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>print("\nOR\n")</div><div>pp_ttable(2, 0b1110)</div></div> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">OR <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0] : r <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">================= <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 : 0 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 : 1 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 0 : 1 <span style="-webkit-text-stroke-width: 0px; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 : 1 <span style="-webkit-text-stroke-width: 0px; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; widows: 2; word-spacing: 0px;"> <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>print("\nXOR\n")</div><div>pp_ttable(2, 0b110)</div></div><span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">XOR <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0] : r <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">================= <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 : 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 : 1 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 0 : 1 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 : 0 <span style="-webkit-text-stroke-width: 0px; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; widows: 2; word-spacing: 0px;"> <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>print("\nAND\n")</div><div>pp_ttable(2, 0b1000)</div></div><span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">AND <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0] : r <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">================= <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 : 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 : 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 0 : 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 : 1 For those three inputs, we can extend the table to show result columns for OR, XOR and then AND, like this:<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>print("\nOR, XOR, then AND result *columns*\n")</div><div>pp_ttable(2, [14, 6, 8])  # A list of results, (in decimal this time)</div></div><span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">OR, XOR, then AND result *columns* <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0] : r[0] r[1] r[2] <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">=========================== <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 : 0 0 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 : 1 1 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 0 : 1 1 0 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 : 1 0 1 <h2 style="text-align: left;">All Truth</h2> Just how many results are possible?Well, i = 2 inputs gives 2**i = 4 possible input boolean combinations; so a result column has 2**i = 4  bits. The number of different result columns is therefore 2**(2**i) = 2**4 = 16We can show all possible results by successive results being a binary count, but this time by column in the results section, (with the LSB being closest to the header row)The pp_table function automatically  generates all possible results if a second parameter of None is used<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>print("\nAll Truths of two inputs!\n")</div><div>pp_ttable(2, None)</div></div><span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">All Truths of two inputs! <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i[1] i[0] : r[0] r[1] r[2] r[3] r[4] r[5] r[6] r[7] r[8] r[9] r[10] r[11] r[12] r[13] r[14] r[15] <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">============================================================================================================== <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 0 : 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 0 1 : 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 0 : 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 <span style="-webkit-text-stroke-width: 0px; color: #2b00fe; font-family: Consolas, "Courier New", monospace; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;"> 1 1 : 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1  We might say that it shows all possible truths for up-to-and-including two inputs. That is because results include outputs not dependent on any or all of those two inputs. For example r[0], and r[15] do not depend on any input as they give constant outputs of 0 and 1, respectively. r[5] is simply ~i[0] and does not depend on i[1]<h3 style="text-align: left;">It's Big, Oh!</h3>The results grow as  2**(2**i), sometimes called double exponential growth! it gets large, quickly!! <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">i | 2**(2**i) <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">--|---------- <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">0 | 2 <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">1 | 4 <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">2 | 16 <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">3 | 256 <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">4 | 65_536 <span style="-webkit-text-stroke-width: 0px; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; word-spacing: 0px;">5 | 4_294_967_296<span style="-webkit-text-stroke-width: 0px; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; text-indent: 0px; text-transform: none; white-space: pre; widows: 2; word-spacing: 0px;"> <h1 style="text-align: left;">The code</h1> Contemplating using AI and needing to get it to understand what I wanted, as well as endless prompting to get it to do what I want, the way I wanted it; I decided on writing it all by myself - I knew I wanted it just so, and the coding would be nothing new to me, just nailing what I wanted to show. <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>"""</div><div>Truthtable prettyprinter</div> <div>Author: Paddy3118  2025/07/19</div> <div># Regions of a truthtable</div> <div>```text</div><div>+----------------- --+---------+</div><div>|      Inputs        | Result  |</div><div>+------------------- +---------+</div><div>| I[2] | I[1] | I[0] |   R     |</div><div>|------|------|------|---------|</div><div>|  0   |  0   |  0   |   0     |</div><div>|  0   |  0   |  1   |   1     |</div><div>|  0   |  1   |  0   |   0     |</div><div>|  0   |  1   |  1   |   1     |</div><div>|  1   |  0   |  0   |   1     |</div><div>|  1   |  0   |  1   |   0     |</div><div>|  1   |  1   |  0   |   1     |</div><div>|  1   |  1   |  1   |   0     |</div><div>+--------------------+---------+</div><div>```</div> <div>"""</div> <div># %%</div><div>def pp_ttable(input_count:int,</div><div>              result_vector: int | list[int] | None,</div><div>              input_vector_name: str='i',</div><div>              result_vector_name: str='r'</div><div>              ) -> None:</div><div>    r_single = r_mult = False</div><div>    match result_vector:</div><div>        case int():     # single result</div><div>            r_single = True</div><div>            r_width = len(f" {result_vector_name}")</div><div>            result_vector = [result_vector]</div><div>        case list():    # multiple results</div><div>            r_mult = True</div><div>            r_width = len(f" {result_vector_name}[{len(result_vector) - 1}]")</div><div>        case None:      # All possible results</div><div>            assert input_count < 4, f"Refuse to write {2**(2**input_count)} result columns."</div><div>            r_mult = True</div><div>            result_vector = list(range(2**(2**input_count)))</div><div>            r_width = len(f" {result_vector_name}[{len(result_vector) - 1}]")</div><div>        case _:</div><div>            raise Typeerror("result must be an int, a list of ints, or None.")</div> <div>    col_width = max(len(f" {input_vector_name}[{input_count - 1}]"), r_width)</div> <div>    rows = []</div><div>    header = []</div><div>    for i in range(input_count-1, -1, -1):</div><div>        col = f"{input_vector_name}[{i:}]"</div><div>        header.append(f"{col:{col_width}}")</div><div>    header.append(': ')</div><div>    if r_single:</div><div>        header.append(f"{result_vector_name:^{col_width}}")</div><div>    else:</div><div>        for i in range(len(result_vector)):</div><div>            col = f"{result_vector_name}[{i:}]"</div><div>            header.append(f"{col:^{col_width}}")</div><div>    rows.append(''.join(header))</div><div>    rows.append('=' * len(rows[-1]))    # header row spacer</div> <div>    result_bits = [f"{res:0{2**input_count}b}"[::-1]</div><div>                    for res in result_vector]</div><div>    </div><div>    for r in range(2**input_count):</div><div>        row = []</div><div>        col_bits = f"{r:0{input_count}b}"</div><div>        for bit in col_bits:</div><div>            row.append(f"{bit:^{col_width}}")</div><div>        </div><div>        row.append(': ')</div> <div>        for res_bits in result_bits:</div><div>            row.append(f"{res_bits[r]:^{col_width}}")</div><div>                </div><div>        rows.append(''.join(row))</div> <div>    print('\n'.join(rows))</div> <div># %%</div><div>print("\nOR\n")</div><div>pp_ttable(2, 0b1110)</div><div>print("\nXOR\n")</div><div>pp_ttable(2, 0b110)</div><div>print("\nAND\n")</div><div>pp_ttable(2, 0b1000)</div><div>    </div> <div># %%</div> <div>print("\nOR, XOR, then AND result *columns*\n")</div><div>pp_ttable(2, [14, 6, 8])  # A list of results, (in decimal this time)</div> <div># %%</div><div>print("\nAll Truths of two inputs!\n")</div><div>pp_ttable(2, None)</div> <div># %%</div> </div> <h4 style="text-align: left;">END. </h4>     Sun, 20 Jul 2025 10:28:00 Incremental combinations without cachinghttp://paddy3118.blogspot.com/feeds/6157910081675774828/comments/default <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiDhcFwRluMUmESYsqTtkwNppqf1z4krD48fhecEBVWqwxG6UN6vaA1CctLiNef3T3kEUip9mQsHGaK6zfHGNo-PICWgj2vo9mFdMU-xSGImhgl8PRc3wgI172uD1R1zRtYCja71vCLWxIZvitL-xTN0cFNHb8jTdTL-JqBYlp50D4CjJuJRuz2" style="margin-left: auto; margin-right: auto;"><img alt="" data-original-height="1024" data-original-width="1024" height="534" src="https://blogger.googleusercontent.com/img/a/AVvXsEiDhcFwRluMUmESYsqTtkwNppqf1z4krD48fhecEBVWqwxG6UN6vaA1CctLiNef3T3kEUip9mQsHGaK6zfHGNo-PICWgj2vo9mFdMU-xSGImhgl8PRc3wgI172uD1R1zRtYCja71vCLWxIZvitL-xTN0cFNHb8jTdTL-JqBYlp50D4CjJuJRuz2=w534-h534" width="534" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Irie server room</td></tr></tbody></table> Someone had <a href="https://discuss.python.org/t/incremental-combination-generation-from-laggy-generators/85052/2" rel="nofollow" target="_blank">a problem</a> where they received initial data d1, worked on all r combinations of the data initially received, but by the time they had finished that, they checked and found there was now extra data d2, and they need to, in total, process the r combinations of all data d1+d2.They don't want to process combinations twice, and the solutions given seemed to generate and store the combinations of  d1, then generate the combinations of d1+d2 but test and reject any combination that was found previously.Seems like a straight-forward answer that is easy to follow, but I thought: Is there a way to create just the extra combinations but without storing all the combinations from before?<h2 style="text-align: left;">My methods</h2>It's out with Vscode as my IDE. I'll be playing with a lot of code that will end up deleted, modified, rerun. I could use a Jupyter notebook, but I can't publish them to my blog satisfactorily. I'll develop a .py file but with cells: a line comment of # %% visually splits the file into cells in the IDE adding buttons and editor commands to execute cells and selected code, in any order, on a restartable kernel running in an interactive window that also runs Ipython.When doodling like this, I often create long lines, spaced to highlight comparisons between other lines. I refactor names to be concise at the time, as what you can take in at a glance helps find patterns. Because of that, this blog post is not written to be read on the small screens of phones. So, its combinations; binomials, nCr.<h2 style="text-align: left;">A start</h2><ul style="text-align: left;"><li>First thing was to try and see patterns in the combinations of d1+d2 minus those of just d1.</li><li>Dealing with sets of values; sets are unordered, so will at some  time need a function to print them in order to aid pattern finding.</li><li>Combinations can be large - use combinations of ints then later work with any type.</li><li>Initial combinations of 0 to n-1 ints I later found to be more awkward to reason about so changed to work with combinations of 1 to n ints. extending to n+x added ints n+1 to n+x to combinations.</li></ul><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div>#!/bin/env python3</div><div>"""</div><div>Check patterns in extending combinations nCr</div><div>"""</div> <div># %%</div><div>from functools import reduce</div><div>from itertools import combinations</div><div>from pprint import pformat</div><div>from typing import Any, Iterable</div> <div># %%</div><div>def nCr(n, r):</div><div>    return combinations(range(1, n+1), r=r)</div> <div>def pf_set(s: set) -> str:</div><div>    "Format set with sorted elements for printing"</div><div>    return  f"{{{pformat(sorted(s), width=120, compact=True)[1:-1]}}}"</div> <div></div></div><h3 style="text-align: left;">Diffs</h3>In the following cell I create combinations for r = 3, and n in some range n_range in c and print successive combinations, and differences between successive combinations.<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div>r = 3</div><div>n_range = range(3, 7)</div><div>print(f"\n# Investigate combinations nCr for {r=}, and {n_range=}\n")</div> <div># generate some combinations</div><div>c = {n: set(nCr(n, r)) for n in n_range}</div> <div>print("c = {")</div><div>for key, val in c.items():</div><div>    print(f" {key:2}: {pf_set(val)},")</div><div>print("    }")</div> <div>def pp_diffsby(c, n_range, r, delta_n):</div><div>    "Print nCr in c diffs by delta_n"</div><div>    print(f"\nDiffs by {delta_n}")</div><div>    all_n = list(n_range)</div><div>    n1 = all_n[0]</div><div>    print(f"      {n1}C{r} = {pf_set(c[n1])}")</div><div>    for n1, n2 in zip(all_n, all_n[delta_n:]):</div><div>        print(f"{n2}C{r} - {n1}C{r} = {pf_set(c[n2] - c[n1])}")</div> <div>pp_diffsby(c, n_range, r, 1)</div><div>pp_diffsby(c, n_range, r, 2)</div> </div><h4 style="text-align: left;">Cell output:</h4><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># Investigate combinations nCr for r=3, and n_range=range(3, 7)</div> <div>c = {</div><div>  3: {(1, 2, 3)},</div><div>  4: {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)},</div><div>  5: {(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 5)},</div><div>  6: {(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6),</div><div> (2, 3, 4), (2, 3, 5), (2, 3, 6), (2, 4, 5), (2, 4, 6), (2, 5, 6), (3, 4, 5), (3, 4, 6), (3, 5, 6), (4, 5, 6)},</div><div>    }</div> <div>Diffs by 1</div><div>      3C3 = {(1, 2, 3)}</div><div>4C3 - 3C3 = {(1, 2, 4), (1, 3, 4), (2, 3, 4)}</div><div>5C3 - 4C3 = {(1, 2, 5), (1, 3, 5), (1, 4, 5), (2, 3, 5), (2, 4, 5), (3, 4, 5)}</div><div>6C3 - 5C3 = {(1, 2, 6), (1, 3, 6), (1, 4, 6), (1, 5, 6), (2, 3, 6), (2, 4, 6), (2, 5, 6), (3, 4, 6), (3, 5, 6), (4, 5, 6)}</div> <div>Diffs by 2</div><div>      3C3 = {(1, 2, 3)}</div><div>5C3 - 3C3 = {(1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 5)}</div><div>6C3 - 4C3 = {(1, 2, 5), (1, 2, 6), (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6), (2, 3, 5), (2, 3, 6), (2, 4, 5),</div><div> (2, 4, 6), (2, 5, 6), (3, 4, 5), (3, 4, 6), (3, 5, 6), (4, 5, 6)}</div> </div><h2 style="text-align: left;">Patterns</h2>Looking at diffs 4C3 - 3C3 each tuple is like they took 3C2 = {(1,2), (1,3), (2,3)} and tagged the extra 4 on to every inner tuple. Lets call this modification extending,5C3 - 4C3 seems to follow the same pattern.<h3 style="text-align: left;">Function extend</h3><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div>def extend(s: set[tuple[Any]], value: Any) -> set[tuple[Any]]:</div><div>    """</div><div>    Returns set of tuples of s with each tuple extended by value</div><div>    """</div><div>    return set((*tpl, value) for tpl in s)</div> <div>s = {(0, 1), (1, 2), (0, 2)}</div><div>print(f"{s = }; {extend(s, 3) = }")</div><div>assert extend(s, 3) == {(0, 1, 3), (1, 2, 3), (0, 2, 3)}</div> <div></div></div><h3 style="text-align: left;">Rename nCr to bino and check extend works</h3>nCr  was originally working with 0..n-1 and bino was 1..n. Now they both do<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div># binomial combinations of ints 1..</div> <div>def bino(n: int, r: int) -> set[tuple[int]]:</div><div>    """</div><div>    All combinations of 1..n ints taken r at a time</div> <div>    bino(4, 3) == {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}</div><div>    """</div><div>    return set(combinations(range(1, n+1), r))</div> <div>print(f"{(bino(4, 3) == (bino(3, 3) | extend(bino(3, 2), value=4)))  =  }")</div> </div><h4 style="text-align: left;">Cell output:</h4><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div style="line-height: 19px;"><div>(bino(4, 3) == (bino(3, 3) | extend(bino(3, 2), value=4)))  =  True</div><div> </div></div></div><h2 style="text-align: left;">Pascal</h2><div>After finding that pattern I went searching for it using Gemini AI. My question was:</div><div style="text-align: left;"><blockquote>show that comb(n+1, r) = comb(n, r) + (n+1)* comb(n, r-1)</blockquote></div><div>The answer said I got my text prompt wrong and mentioned <a href="https://en.wikipedia.org/wiki/Pascal%27s_rule" rel="nofollow" target="_blank">Pascals Rule</a>.</div><div>I scanned the page, as I no time for how things were expressed but it seemed reasonable that I had the algorithm right, and that there were relations of some kind for bigger differences in n.</div><div> </div><h3 style="text-align: left;">Pascals rule checker</h3><div>I wrote a function to do the check then exercised it  (a few times, now deleted).</div><div> </div><div><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div><div>def pascals_rule(n: int, r: int) -> bool:</div><div>    "check C(n+1, r) == C(n, r) | extend(C(n, r-1), n + 1)"</div><div>    return bino(n + 1, r) == bino(n, r) | extend(bino(n, r - 1), n + 1)</div> <div>assert pascals_rule(6, 3)</div></div></div><div> </div><h2 style="text-align: left;">Diff by 1 extra item "done", attempting diff by 2.</h2><div>Looking back at the diffs by 2 table and looking for patterns I thought I might need different types of extension functions modifying the tuples within sets in different ways - it seemed "mathematical" so...</div><div> </div><div><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div># Some functions that may be needed</div> <div>def extend_mul(s: set[tuple[Any]], e: Iterable) -> set[tuple[Any]]:</div><div>    """</div><div>    set where each tuple of s is extended in turn, by every item in e</div> <div>    s = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}</div><div>    extend_mul(s, (5, 6)) == {(1, 2, 3, 5), (1, 2, 3, 6),</div><div>                              (1, 2, 4, 5), (1, 2, 4, 6),</div><div>                              (1, 3, 4, 5), (1, 3, 4, 6),</div><div>                              (2, 3, 4, 5), (2, 3, 4, 6)}</div><div>    """</div><div>    return {(*t, item) for t in s for item in e}</div> <div>def extend_muli(s: set[tuple[Any]], e: Iterable) -> set[tuple[Any]]:</div><div>    """</div><div>    set where each tuple of s is extended in turn, by every *item in e</div> <div>    s = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}</div><div>    extend_muli(s, ((5,), (6,))) == {(1, 2, 3, 5), (1, 2, 3, 6),</div><div>                              (1, 2, 4, 5), (1, 2, 4, 6),</div><div>                              (1, 3, 4, 5), (1, 3, 4, 6),</div><div>                              (2, 3, 4, 5), (2, 3, 4, 6)}</div><div>    """</div><div>    return {(*t, *item) for t in s for item in e}</div> <div>def extend_add(s: set[tuple[Any]], e: Iterable) -> set[tuple[Any]]:</div><div>    """</div><div>    set where each tuple of s is extended once, all  items of *e</div> <div>    s = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}</div><div>    extend_add(s, (5, 6)) == {(1, 2, 3, 5, 6), (1, 2, 4, 5, 6),</div><div>                              (1, 3, 4, 5, 6), (2, 3, 4, 5, 6)}</div><div>    """</div><div>    return {(*t, *e) for t in s}</div> </div></div><div> </div><h2 style="text-align: left;">Diff by 2 pattern finding</h2>It was incremental - find a pattern in 3C?, subtract it from 5C3, Find a pattern in 3C? that covers part of the remainder; repeat. (Where ? <=3).I also shortened function pf_set to pf so it would take less space when printing formatted expressions in f-strings<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div>print(f"To simplify formatted printing of sets: {(pf:=pf_set) = }\n")</div> <div>print("# Looking again at the diffs by 2 i.e. `5C3 - 3C3`")</div><div>n, r, x = 3, 3, 2</div> <div>print(f"5C3 - 3C3 = {pf(bino(5, 3) - bino(3, 3))}")</div><div>print(f"\n  There's {pf(bino(3, 3-1)) = } in there with each tuple extended by 4 and by 5 ")</div><div>print(f"\n  {pf(extend_mul(bino(3, 3-1), (3+1, 3+2))) = }")</div><div>print(f"\n  Whats left: {pf(tmp1 := (bino(5, 3) - bino(3, 3) - extend_mul(bino(3, 3-1), (3+1, 3+2)))) = }")</div><div>print(f"\n    Now {pf(bino(3, 3-2)) = }")</div><div>print(f"\n    So {pf(tmp2 := (extend_add(bino(3, 3-2), (3+1, 3+2)))) = }")</div><div>print(f"\n  Finally: {pf(tmp1 - tmp2) = }")</div></div><h4 style="text-align: left;">Cell output:</h4><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div>To simplify formatted printing of sets: (pf:=pf_set) = <function pf_set at 0x7f1bd45365c0></div> <div># Looking again at the diffs by 2 i.e. `5C3 - 3C3`</div><div>5C3 - 3C3 = {(1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 5)}</div> <div>  There's pf(bino(3, 3-1)) = '{(1, 2), (1, 3), (2, 3)}' in there with each tuple extended by 4 and by 5 </div> <div>  pf(extend_mul(bino(3, 3-1), (3+1, 3+2))) = '{(1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (2, 3, 4), (2, 3, 5)}'</div> <div>  Whats left: pf(tmp1 := (bino(5, 3) - bino(3, 3) - extend_mul(bino(3, 3-1), (3+1, 3+2)))) = '{(1, 4, 5), (2, 4, 5), (3, 4, 5)}'</div> <div>    Now pf(bino(3, 3-2)) = '{(1,), (2,), (3,)}'</div> <div>    So pf(tmp2 := (extend_add(bino(3, 3-2), (3+1, 3+2)))) = '{(1, 4, 5), (2, 4, 5), (3, 4, 5)}'</div> <div>  Finally: pf(tmp1 - tmp2) = '{}'</div></div><h3 style="text-align: left;">Behold Diff by 2</h3><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div><div>n, r, x = None, None, None</div> <div>print(f"\n# lets set some variables and use those for the diffs by 2")</div><div>print(f"  {(n:=3), (r:=3) = }")</div><div>print(f"  {len(bino(n+2, r)) = }")</div><div>print(f"  {bino(n+2, r) == (bino(n, r) | extend_mul(bino(n, r-1), (n+1, n+2)) | extend_add(bino(n, r-2), (n+1, n+2))) = }")</div></div><h4 style="text-align: left;">Cell output:</h4><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># lets set some variables and use those for the diffs by 2</div><div>  (n:=3), (r:=3) = (3, 3)</div><div>  len(bino(n+2, r)) = 10</div><div>  bino(n+2, r) == (bino(n, r) | extend_mul(bino(n, r-1), (n+1, n+2)) | extend_add(bino(n, r-2), (n+1, n+2))) = True</div></div><h2 style="text-align: left;">Pascals rules for increasing diffs</h2>I followed the same method for diffs of three and ended up with these three functions:<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div># By similar observation and much checking:</div><div>def pascal_rule_1(n, r) -> bool:</div><div>    """</div><div>    Checks bino(n+1, r) == (bino(n, r)</div><div>                            | extend_add(bino(n, r-1), (n+1,)))</div><div>    """</div><div>    return bino(n+1, r) == (bino(n, r)</div><div>                            | extend_add(bino(n, r-1), (n+1,)))</div> <div>def pascal_rule_2(n, r) -> bool:</div><div>    """</div><div>    Checks bino(n+2, r) == (bino(n, r)</div><div>                            | extend_mul(bino(n, r-1), (n+1, n+2))</div><div>                            | extend_add(bino(n, r-2), (n+1, n+2)))</div><div>    """</div><div>    return bino(n+2, r) == (bino(n, r)</div><div>                            | extend_mul(bino(n, r-1), (n+1, n+2))</div><div>                            | extend_add(bino(n, r-2), (n+1, n+2)))</div> <div>def pascal_rule_3(n, r) -> bool:</div><div>    """</div><div>    Checks bino(n+3, r) == (bino(n, r)</div><div>                            | extend_muli(bino(n, r-1), tuple(combinations((n+1, n+2, n+3), 1)))</div><div>                            | extend_muli(bino(n, r-2), tuple(combinations((n+1, n+2, n+3), 2)))</div><div>                            | extend_muli(bino(n, r-3), tuple(combinations((n+1, n+2, n+3), 3)))</div><div>                            )</div><div>    """</div><div>    extra_n = tuple(range(n+1, n+4))  # n..n+3 inclusive</div><div>    return bino(n+3, r) == (bino(n, r)</div><div>                            | extend_muli(bino(n, r-1), tuple(combinations(extra_n, 1)))</div><div>                            | extend_muli(bino(n, r-2), tuple(combinations(extra_n, 2)))</div><div>                            | extend_muli(bino(n, r-3), tuple(combinations(extra_n, 3)))</div><div>                            )</div> <div># %%</div><div># Simple Checks</div><div>assert pascal_rule_1(7, 4)</div><div>assert pascal_rule_2(9, 4)</div><div>assert pascal_rule_3(11, 4)</div> </div><h3 style="text-align: left;">Generalised Pascals rule</h3>What can I say, I looked for patterns in the pascals rule functions for discrete diffs and tried to find patterns. I looked deeper into identities between the extend functions.I finally found the following function that passed my tests, (many not shown). <div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div><div># from pascal_rule_3 to pascal_rule_x</div> <div>def pascal_rule_x(n, r, x) -> bool:</div><div>    """</div><div>    Checks bino(n+x, r) == union_reduce(<extend_muli selections of bino(n, 0 < r < r)> )</div> <div>    ie if already used bino(n, r) and along comes x more for n then it shows how to</div><div>    calculate bino(n+x, r) without the need for storing bino(n, r)</div> <div>    """</div><div>    extra_n = tuple(range(n+1, n+x+1))  # n..n+x inclusive</div><div>    n_r_terms = (extend_muli((bino(n, r-i) if r-i > 0 else {()}),   # extend this</div><div>                             tuple(combinations(extra_n, i)))       # by this</div><div>                for i in range(min(x, r) + 1))</div><div>    reduction = reduce(set.union, n_r_terms, set())</div><div>    return bino(n+x, r) == reduction</div> <div>assert pascal_rule_x(11, 4, 3)  # n, r, x</div><div>assert pascal_rule_x(11, 5, 4)</div><div>assert pascal_rule_x(3, 2, 3)</div></div>I don't like the if r-i > 0 else {()} bit as it doesn't seem elegant. There is probably some identity to be found that would make it disappear but, you know.<h1 style="text-align: left;">Back to the original problem</h1>If comb(d1, r) is processed and then we find an extra d2 items, then we want to process extra_comb(d1, r, d2) where extra_comb does not include or save comb of d1.We just need to exclude the nCr term in reduction of function pascal_rule_x.Eventually I arrive at<div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div>first = list(combinations('abcd', 3))</div><div>first</div><div># %%</div><div>all = list(combinations('abcdef', 3))</div><div># %%</div><div>extra = sorted(set(all) - set(first))</div><div>extra</div><div># %%</div> <div>def extra_combs(orig='abcd', r=3, delta='ef'):</div><div>    C = combinations  # Less typing</div><div>    extra_n = tuple(delta)  # n..n+x inclusive</div><div>    n = tuple(orig)</div><div>    n_r_terms = (extend_muli((C(n, r-i) if r-i > 0 else {()}),   # extend this</div><div>                             tuple(C(extra_n, i)))               # by this</div><div>                for i in range(1, min(len(extra_n), r) + 1))     # miss C(n, r)</div><div>    reduction = reduce(set.union, n_r_terms, set())</div><div>    # set(C(n+extra_n, r)) - reduction == set(C(n, r))</div><div>    return reduction</div> <div>n, r, delta = 'abcd', 3, 'efg'</div><div>assert set(combinations(n+delta, r)) \</div><div>         == set(combinations(n, r)).union(extra_combs(n, r, delta))</div></div><h2 style="text-align: left;">Tests</h2><div style="background-color: #1f1f1f; color: #cccccc; font-family: Consolas, "Courier New", monospace; font-size: 14px; line-height: 19px; white-space: pre;"><div># %%</div> <div># Test comb(n+x, r) == comb(n, r) | extra_combs(n, r, x)</div> <div>n, r, delta_ = 'abcdefg', 3, 'hijkl'</div> <div>for r in range(len(n)):</div><div>    for delta in (delta_[:i] for i in range(len(delta_))):</div><div>        combnx = set(combinations(n+delta, r))</div><div>        combn = set(combinations(n, r))</div><div>        extra = extra_combs(n, r, delta)</div><div>        assert combnx == (combn | extra), f"Whoops! For  {(n, r, delta) = }"</div><div>        # checks that extra does not generate any of combn</div><div>        assert not extra.intersection(combn), f"Whoops! For  {(n, r, delta) = }"</div> </div> END. Sun, 23 Mar 2025 08:54:00 UTC