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Go deh!Mainly Tech projects on Python and Electronic Design Automation.Sparse Rangeshttp://paddy3118.blogspot.com/feeds/4876301501222443352/comments/default <div _ngcontent-ng-c3428423599="" aria-busy="false" aria-live="polite" class="markdown markdown-main-panel enable-luminous-fast-follows enable-updated-hr-color tutor-markdown-rendering" dir="ltr" id="model-response-message-contentr_62ca493757305bf2" style="--animation-duration: 400ms; --fade-animation-function: ease-out; font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><h3 data-path-to-node="2" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">Implementing <code data-index-in-node="13" data-path-to-node="2" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range</code>: From a Python Discuss Idea to a Sieve Stress Test</h3>I found an interesting thread over on the Python Discuss forum titled "<a href="https://discuss.python.org/t/possibility-to-exclude-ranges-from-range/107885" rel="nofollow" target="_blank">Possibility to exclude ranges from range</a>". The initial discussion revolved around a common (?), developer need of : how to cleanly skip specific blocks of ints in a range of ints without writing clunky, nested <code data-index-in-node="285" data-path-to-node="3" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">if/continue</code> logic, and without doing something memory-heavy like casting everything into a massive <code data-index-in-node="384" data-path-to-node="3" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">set</code>.The consensus was that Python's native <code data-index-in-node="39" data-path-to-node="4" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">range</code> is beautiful because it’s a lightweight, memory-efficient arithmetic engine. It doesn't store numbers in RAM; it just calculates them on the fly. But the moment you need to punch holes in it—say, "give me all numbers from 1 to 1000, except for multiples of 5, and except for the block from 200 to 300"—you lose that structural elegance.This <a href="https://discuss.python.org/t/possibility-to-exclude-ranges-from-range/107885/46" rel="nofollow" target="_blank">inspired me</a> to build a clean, production-grade abstraction to solve this kind of thng: <code data-index-in-node="77" data-path-to-node="5" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range</code>.<h4 data-path-to-node="6" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">The Architectural Concept</h4>The core idea began with a simple mathematical definition: a Sparse range instance is created by a set of (python) ranges S_in that hold possible output values and a set of ranges S_ex that exclude possible output values. A Sparse_range instance can be called with a start, stop, and step argument that generate trial output candidates that are checked against S_in and S_ex to see if they are allowed/filtered. An item is eligible for the final output if its value is present in at least one of our permitted baseline ranges S_in, unless that value happens to be intercepted by one of our forbidden exclusion ranges S_ex. On top of that structure, the outer <code data-index-in-node="314" data-path-to-node="7" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range</code> wrapper behaves like a native Python range—it accepts its own outer <code data-index-in-node="403" data-path-to-node="7" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">start</code>, <code data-index-in-node="410" data-path-to-node="7" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">stop</code>, and <code data-index-in-node="420" data-path-to-node="7" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">step</code> constraints, matching the target window requested by the user.To make the evaluation stateless and fast(?), the engine must evaluate values on the fly. However, native Python <code data-index-in-node="111" data-path-to-node="8" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">range</code> objects are immutable and opaque; they don't natively maintain operational iteration states, nor do they know how to coordinate with one another.To bridge this gap, the engine internally upgrades every raw Python range passed to collections S_in and S_ex into custom <code data-index-in-node="88" data-path-to-node="9" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">RangePointers</code>.<div _ngcontent-ng-c2702060661="" class="code-block ng-tns-c2702060661-246 ng-animate-disabled ng-trigger ng-trigger-codeBlockRevealAnimation" data-hveid="0" data-ved="0CAAQhtANahgKEwi4k8qG4aaVAxUAAAAAHQAAAAAQ_wQ" style="display: block; font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="formatted-code-block-internal-container ng-tns-c2702060661-246" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-246" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="code-block-decoration header-formatted gds-emphasized-body-m ng-tns-c2702060661-246 ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> <div style="background-color: #121314; color: #bbbebf; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div> +----------------------------------------+</div><div>               |  Outer Loop: start, stop, step         |</div><div>               +----------------------------------------+</div><div>                                   |</div><div>                  [ Evaluates current candidate value ]</div><div>                                   |</div><div>                                   v</div><div>         +--------------------------------------------------+</div><div>         | Is value in any S_in? AND NOT in any S_ex?       |</div><div>         +--------------------------------------------------+</div><div>             |                                          |</div><div>    (Using forward-aligned                     (Using backward-aligned</div><div>     RangePointers for step > 0)                RangePointers for step < 0)</div><div> </div></div><div _ngcontent-ng-c2702060661="" class="buttons ng-tns-c2702060661-246 ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><button _ngcontent-ng-c442952589="" aria-label="Download code" class="mdc-icon-button mat-mdc-icon-button mat-mdc-button-base mat-badge mat-unthemed mat-badge-overlap mat-badge-above mat-badge-after mat-badge-small mat-badge-hidden ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"></button><button _ngcontent-ng-c442952589="" aria-label="Copy code" class="mdc-icon-button mat-mdc-icon-button mat-mdc-button-base mat-badge mat-unthemed mat-badge-overlap mat-badge-above mat-badge-after mat-badge-small mat-badge-hidden ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"></button></div></div><pre _ngcontent-ng-c2702060661="" class="ng-tns-c2702060661-246" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><code _ngcontent-ng-c2702060661="" class="code-container formatted ng-tns-c2702060661-246" data-test-id="code-content" role="text" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> </code></pre></div></div></div><h4 data-path-to-node="11" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">Why We Need <code data-index-in-node="12" data-path-to-node="11" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">RangePointer</code> and Direction Alignment</h4>When the outer loop steps through values, the internal component ranges must follow along. If the outer loop is counting upwards (<code data-index-in-node="130" data-path-to-node="12" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">step > 0</code>), all internal evaluation cursors must track upwards. If the outer loop reverses direction and counts downwards (<code data-index-in-node="252" data-path-to-node="12" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">step < 0</code>), every single internal range must generate the same values, but in the reverse order, and without using up extra memory for storing values.If they don't match directions, the leapfrogging logic breaks. A forward-running pointer cannot efficiently tell a backward-running outer loop where the next valid alignment boundary is without wasting CPU cycles scanning dead space or whatever.Therefore, when a <code data-index-in-node="18" data-path-to-node="14" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range</code> instance is invoked, the engine inspects the outer <code data-index-in-node="73" data-path-to-node="14" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">step</code> direction and enforces a unified traversal direction across every single internal S_in and S_ex sequence. If an internal range's native direction opposes the outer loop, it must be completely mathematically inverted.<h4 data-path-to-node="15" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">The Range Reversal Formula</h4>Reversing an arithmetic progression isn't as simple as swapping the start and stop boundaries. Because Python ranges stop before reaching the termination boundary, reversing a range requires recalculating its new alignment based on its exact step constraints.To reverse a range mathematically, the engine computes:<ol data-path-to-node="18" start="1" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important; padding-inline-start: 32px;"><li style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">New Step: The step sign is flipped (<code data-index-in-node="36" data-path-to-node="18,0,0" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">-step</code>).</li><li style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">New Start: The final valid item generated by the original range becomes the new starting point. This is computed using the remainder of the length of the sequence:<div data-path-to-node="18,1,1" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div class="math-block" data-math="\text{new\_start} = \text{start} + (\text{length} - 1) \times \text{step}" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">new_start = start + (length - 1) × step</div><div class="math-block" data-math="\text{new\_start} = \text{start} + (\text{length} - 1) \times \text{step}" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> </div></div></li><li style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">New Stop: The original <code data-index-in-node="23" data-path-to-node="18,2,0" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">start</code> boundary shifts out by one step inverted to act as the non-inclusive terminal wall:<div data-path-to-node="18,2,1" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div class="math-block" data-math="\text{new\_stop} = \text{start} - \text{step}" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">new_stop = start - step</div></div></li></ol>By normalizing all <code data-index-in-node="19" data-path-to-node="19" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">RangePointer</code> instances to track in the exact same direction as the outer loop, the engine can efficiently step through candidate numbers, skipping blocks of excluded data in O(1) or O(K) time (where K is the number of active ranges), keeping the memory footprint at absolute zero.<h4 data-path-to-node="19" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important; text-align: left;">The code</h4>sparse_range.py <div style="background-color: #121314; color: #bbbebf; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>#!/bin/env python3</div><div># Author: Donald "Paddy" McCarthy. paddy3118@gmail.com</div><div># 27/06/2026</div> <div>#%%</div><div>"""</div><div>sparse_range - Reusable Sparse Integer Generators</div><div>===================================================</div> <div>This module provides a configured factory class `sparse_range`. You define your </div><div>global permitted (`s_in`) and forbidden (`s_ex`) integer lists once, and the </div><div>resulting object can be called repeatedly with customized window configurations </div><div>exactly like Python's built-in `range()` function.</div> <div>Usage</div><div>-----</div><div>    >>> from sparse_range import sparse_range</div><div>    >>> # Configure the template rules</div><div>    >>> my_filter = sparse_range(s_in=[range(0, 100, 2)], s_ex=[range(10, 20, 3)])</div><div>    >>></div><div>    >>> # Call it like the built-in range function</div><div>    >>> sequence = my_filter(0, 30, 3)</div><div>    >>> list(sequence)</div><div>    [0, 6, 24]</div> <div>Command Line Interface</div><div>----------------------</div><div>Run this script directly to run differential limits tests or inspect help files:</div><div>    python sparse_range.py</div><div>"""</div> <div>import sys</div><div>from typing import List, Optional, Iterator, Set, Dict</div> <div># Explicitly define module exports</div><div>__all__ = ['sparse_range']</div> <div>class RangePointer:</div><div>    """</div><div>    Manages a single sub-range's mathematical cursor.</div><div>    Dynamically aligns itself relative to the iteration direction of the caller.</div><div>    """</div><div>    def __init__(self, r: range) -> None:</div><div>        """Initializes structural constraints of a single tracking range cursor."""</div><div>        self.start: int = r.start</div><div>        self.stop: int = r.stop</div><div>        self.step: int = r.step</div><div>        self.length: int = len(r)</div><div>        </div><div>        # Calculate the actual last element generated by this range</div><div>        if self.length > 0:</div><div>            self.last: Optional[int] = r.start + (self.length - 1) * r.step</div><div>        else:</div><div>            self.last = None</div> <div>    def align_to_or_past(self, target: int, direction: int) -> Optional[int]:</div><div>        """</div><div>        Leapfrogs the cursor instantly to the closest valid element </div><div>        aligned with the runtime's traversal direction.</div><div>        """</div><div>        if self.length == 0 or self.last is None:</div><div>            return None</div> <div>        # 1. Check if the target is entirely out of bounds for this sub-range</div><div>        low, high = min(self.start, self.last), max(self.start, self.last)</div><div>        if target < low:</div><div>            return self.start if direction > 0 else None</div><div>        if target > high:</div><div>            return None if direction > 0 else self.last</div> <div>        # 2. Project target onto the arithmetic sequence of this range</div><div>        distance = target - self.start</div><div>        </div><div>        if self.step > 0:</div><div>            k = (distance + self.step - 1) // self.step if distance > 0 else distance // self.step</div><div>        else:</div><div>            k = distance // self.step if distance > 0 else (distance + self.step + 1) // self.step</div><div>            </div><div>        candidate = self.start + k * self.step</div> <div>        # 3. Sweep forward/backward based on global runtime direction</div><div>        if direction > 0:</div><div>            while candidate < target:</div><div>                candidate += self.step</div><div>            if low <= candidate <= high:</div><div>                return candidate</div><div>        else:</div><div>            while candidate > target:</div><div>                candidate -= self.step</div><div>            if low <= candidate <= high:</div><div>                return candidate</div><div>                </div><div>        return None</div> <div>class SparseRangeIterable:</div><div>    """The execution runner returned when a configured sparse_range is called."""</div><div>    def __init__(self, start: int, stop: int, step: int, s_in: List[range], s_ex: List[range]) -> None:</div><div>        """Prepares runtime matrices for specific window iterations."""</div><div>        if step == 0:</div><div>            raise ValueError("sparse_range loop step argument must not be zero")</div><div>        self.start: int = start</div><div>        self.stop: int = stop</div><div>        self.step: int = step</div><div>        self.direction: int = 1 if step > 0 else -1</div><div>        </div><div>        self.in_ranges: List[RangePointer] = [RangePointer(r) for r in s_in if len(r) > 0]</div><div>        self.ex_ranges: List[RangePointer] = [RangePointer(r) for r in s_ex if len(r) > 0]</div> <div>    def __iter__(self) -> Iterator[int]:</div><div>        """Iterates through matching sparse elements using double-pointer acceleration."""</div><div>        target = self.start</div><div>        stop = self.stop</div><div>        step = self.step</div><div>        dir_mask = self.direction</div> <div>        while (dir_mask > 0 and target < stop) or (dir_mask < 0 and target > stop):</div><div>            # Align all input sub-ranges and collect their active values</div><div>            active_in: Dict[RangePointer, int] = {}</div><div>            for p in self.in_ranges:</div><div>                val = p.align_to_or_past(target, dir_mask)</div><div>                if val is not None:</div><div>                    active_in[p] = val</div> <div>            if not active_in:</div><div>                break</div> <div>            # Align all exclusion sub-ranges</div><div>            active_ex: Dict[RangePointer, int] = {}</div><div>            for p in self.ex_ranges:</div><div>                val = p.align_to_or_past(target, dir_mask)</div><div>                if val is not None:</div><div>                    active_ex[p] = val</div> <div>            # Dynamic Execution Strategy Optimization Lookup</div><div>            is_included = False</div><div>            if len(active_in) <= len(active_ex):</div><div>                if any(val == target for val in active_in.values()):</div><div>                    if not any(val == target for val in active_ex.values()):</div><div>                        is_included = True</div><div>            else:</div><div>                if not any(val == target for val in active_ex.values()):</div><div>                    if any(val == target for val in active_in.values()):</div><div>                        is_included = True</div> <div>            if is_included:</div><div>                yield target</div><div>            </div><div>            target += step</div> <div>class sparse_range:</div><div>    """</div><div>    A factory pattern configuration setup that caches structural filter limits.</div><div>    Instances can be directly invoked exactly like the standard range() construct.</div><div>    """</div><div>    def __init__(self, s_in: Optional[List[range]] = None, s_ex: Optional[List[range]] = None) -> None:</div><div>        """Instantiates the filtering template for subsequent executions."""</div><div>        self._s_in_raw: List[range] = s_in or []</div><div>        self._s_ex_raw: List[range] = s_ex or []</div> <div>    def __repr__(self) -> str:</div><div>        """Returns structural serialization details for troubleshooting."""</div><div>        return f"sparse_range(s_in={self._s_in_raw}, s_ex={self._s_ex_raw})"</div> <div>    def __call__(self, start: int, stop: Optional[int] = None, step: int = 1) -> SparseRangeIterable:</div><div>        """Mimics the signature properties of Python's default range() function."""</div><div>        if stop is None:</div><div>            stop = start</div><div>            start = 0</div><div>        return SparseRangeIterable(start, stop, step, self._s_in_raw, self._s_ex_raw)</div> <div>def _expensive_sparse_range(start: int, stop: int, step: int, s_in: List[range], s_ex: List[range]) -> List[int]:</div><div>    """Reference high-memory execution harness to generate baseline validation benchmarks."""</div><div>    allowed: Set[int] = set()</div><div>    for r in (s_in or []):</div><div>        allowed.update(r)</div><div>    for r in (s_ex or []):</div><div>        allowed.difference_update(r)</div><div>        </div><div>    target = start</div><div>    direction = 1 if step > 0 else -1</div><div>    result: List[int] = []</div><div>    while (direction > 0 and target < stop) or (direction < 0 and target > stop):</div><div>        if target in allowed:</div><div>            result.append(target)</div><div>        target += step</div><div>    return result</div> <div>def run_tests() -> None:</div><div>    """Runs structural verification configurations spanning cross-boundary constraints."""</div><div>    print("Running Factory Pattern Limits Verification Suite (-1000 to +1000)...\n")</div><div>    </div><div>    # 1. Main dynamic test setup</div><div>    filter_config = sparse_range(</div><div>        s_in=[range(-500, 500, 3)],</div><div>        s_ex=[range(-50, 50, 2), range(200, 300, 5)]</div><div>    )</div><div>    </div><div>    test_windows = [</div><div>        ("Standard positive crawl", (0, 200, 2)),</div><div>        ("Negative direction crawl", (400, -400, -5)),</div><div>        ("Large step slice jump", (-1000, 1000, 25)),</div><div>        ("Completely out of bounds window", (600, 900, 1))</div><div>    ]</div><div>    </div><div>    # 2. Explicit Empty Input Edge Case Verification Setup</div><div>    empty_filter_config = sparse_range(s_in=[range(0, 0), range(10, 5)], s_ex=[range(0, 100)])</div><div>    test_windows.append(("Empty S_in Edge Case", (0, 50, 1)))</div><div>    </div><div>    passed = 0</div><div>    for name, args in test_windows:</div><div>        start, stop, step = args</div><div>        </div><div>        # Check which config to evaluate against</div><div>        cfg = empty_filter_config if "Empty" in name else filter_config</div><div>        </div><div>        actual_output = list(cfg(start, stop, step))</div><div>        expected_output = _expensive_sparse_range(start, stop, step, cfg._s_in_raw, cfg._s_ex_raw)</div><div>        </div><div>        if actual_output == expected_output:</div><div>            print(f"✅ PASS: {name:<35} -> args: ({start}, {stop}, {step})")</div><div>            passed += 1</div><div>        else:</div><div>            print(f"❌ FAIL: {name:<35}")</div><div>            print(f"   Expected: {expected_output}")</div><div>            print(f"   Got:      {actual_output}")</div><div>            </div><div>    print(f"\nTest Summary: {passed}/{len(test_windows)} passed successfully.")</div> <div>if __name__ == "__main__":</div><div>    args = sys.argv[1:]</div><div>    if not args or "--test" in args:</div><div>        run_tests()</div><div>    elif "--help" in args or "-h" in args:</div><div>        print("""sparse_range Factory Engine\n---------------------------""")</div><div>    elif "--doc" in args:</div><div>        print(__doc__.strip())</div> </div> A run of included tests Running Factory Pattern Limits Verification Suite (-1000 to +1000)... ✅ PASS: Standard positive crawl             -> args: (0, 200, 2) ✅ PASS: Negative direction crawl            -> args: (400, -400, -5) ✅ PASS: Large step slice jump               -> args: (-1000, 1000, 25) ✅ PASS: Completely out of bounds window     -> args: (600, 900, 1) ✅ PASS: Empty S_in Edge Case                -> args: (0, 50, 1) Test Summary: 5/5 passed successfully. <h3 data-path-to-node="20" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important; text-align: left;">The Ultimate Stress Test: A Declarative Sieve</h3>It's easy to write a basic range filter that skips a static block of numbers. But to prove that this <code data-index-in-node="101" data-path-to-node="21" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">RangePointer</code> inversion and alignment math is completely airtight across dozens of concurrent boundaries, I decided to implement a completely declarative Sieve of Eratosthenes.Instead of allocating a large mutable array of booleans in memory and procedurally striking out composites, we can use pure sequence logic:<ol data-path-to-node="23" start="1" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important; padding-inline-start: 32px;"><li style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">The Permitted Range S_in: Our baseline pool of candidate of ints starting at 2: <code data-index-in-node="85" data-path-to-node="23,0,0" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">[range(2, limit + 1)]</code>.</li><li style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">The Exclusion ranges S_ex: A list of independent arithmetic progression ranges tracking the composites for every factor m up to √(limit).</li></ol>Each exclusion range/stream starts at m × m (instead of m × 2). Any composite smaller than m × m  has a prime factor smaller than m, meaning it has already been intercepted by an earlier stream. For example by the time m=5 runs, numbers like 10, 15, and 20 are already masked out by the ranges/streams for 2 and 3. The first unseen composite is 5 × 5 = 25.<h4 data-path-to-node="25" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">Putting it to the Test</h4>Although the module sparse_range.py has its own tests, to further verify the integrity of the design I put together a test module called <code data-index-in-node="75" data-path-to-node="26" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range_sieve_test.py</code>. It packages the sieve creation into an isolated factory function and validates the output against a traditional, trusted array sieve across both a forward sweep and a reversed backward pass.<div _ngcontent-ng-c2702060661="" class="code-block ng-tns-c2702060661-247 ng-animate-disabled ng-trigger ng-trigger-codeBlockRevealAnimation" data-hveid="0" data-ved="0CAAQhtANahgKEwi4k8qG4aaVAxUAAAAAHQAAAAAQgAU" style="display: block; font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="formatted-code-block-internal-container ng-tns-c2702060661-247" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-247" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">sparse_range_sieve_test.py</div><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-247" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> </div><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-247" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> <div style="background-color: #121314; color: #bbbebf; font-family: Consolas, 'Courier New', monospace; font-size: 14px; font-weight: normal; line-height: 19px; white-space: pre;"><div>#!/bin/env python3</div><div># Author: Donald "Paddy" McCarthy. paddy3118@gmail.com</div><div># 27/06/2026</div> <div>#%%</div><div>"""</div><div>sparse_range_sieve_test - Prime Generation Performance Test Harness</div><div>===================================================================</div> <div>This module verifies the mathematical integrity of the `sparse_range` tracking</div><div>engine by utilizing it to implement a declarative Sieve of Eratosthenes. </div><div>It tests overlapping cross-boundary constraints by instantiating dozens of </div><div>simultaneous exclusion ranges running with diverse step intervals.</div><div>"""</div> <div>from typing import List</div><div>from sparse_range import sparse_range</div> <div>def traditional_sieve(n: int) -> List[int]:</div><div>    """Generates prime numbers up to N using a standard boolean array sieve."""</div><div>    if n < 2:</div><div>        return []</div><div>    sieve = [True] * (n + 1)</div><div>    sieve[0] = sieve[1] = False</div><div>    </div><div>    for p in range(2, int(n**0.5) + 1):</div><div>        if sieve[p]:</div><div>            for i in range(p * p, n + 1, p):</div><div>                sieve[i] = False</div><div>                </div><div>    return [i for i, is_prime in enumerate(sieve) if is_prime]</div> <div>def build_prime_sieve(limit: int) -> sparse_range:</div><div>    """</div><div>    Constructs a sparse_range configured as a declarative Sieve of Eratosthenes.</div><div>    </div><div>    How S_in and S_ex constitute a "sieved" range:</div><div>    ----------------------------------------------</div><div>    Instead of tracking a mutable array of booleans in memory and crossing off</div><div>    composites procedurally, this factory models the Sieve of Eratosthenes </div><div>    declaratively using pure sequence logic.</div><div>    </div><div>    1. The Permitted Range (S_in):</div><div>       We start with the baseline domain of all potential prime candidates. Since</div><div>       1 is not prime, our domain is a single continuous range starting at 2:</div><div>       S_in = [range(2, limit + 1)]</div><div>       </div><div>    2. The Exclusion Streams (S_ex):</div><div>       To eliminate non-primes, we generate an independent exclusion range for </div><div>       every integer 'm' from 2 up to sqrt(limit). Each range tracks the periodic </div><div>       sequence of composites generated by that factor.</div><div>       </div><div>       - Step factor: The range increments by 'm' (e.g., for m=3, it targets </div><div>         every 3rd number).</div><div>       - Optimization (m * m): Rather than starting at m * 2, each exclusion stream </div><div>         starts at m * m. Any composite number smaller than m * m must possess a </div><div>         prime factor smaller than 'm', meaning it is already guaranteed to be </div><div>         intercepted by a previous exclusion stream.</div><div>         </div><div>    When the resulting factory is iterated, the sparse_range engine dynamically</div><div>    clashes these arithmetic progressions together, masking out composites instantly</div><div>    without storing any tables in RAM.</div><div>    """</div><div>    # Baseline candidate space</div><div>    s_in: List[range] = [range(2, limit + 1)]</div><div>    </div><div>    # Composite elimination progressions</div><div>    s_ex: List[range] = [</div><div>        range(m * m, limit + 1, m) </div><div>        for m in range(2, int((limit + 1)**0.5) + 1)</div><div>    ]</div><div>    </div><div>    return sparse_range(s_in=s_in, s_ex=s_ex)</div> <div>def run_sieve_tests(limit: int = 200) -> None:</div><div>    """Executes forward and backward verification passes against traditional algorithms."""</div><div>    print(f"Beginning Sparse Range Sieve Evaluation Suite (N = {limit})...")</div><div>    print("----------------------------------------------------------------")</div> <div>    # 1. Generate baseline ground-truth primes</div><div>    expected_forward: List[int] = traditional_sieve(limit)</div><div>    expected_backward: List[int] = list(reversed(expected_forward))</div> <div>    # 2. Build the sieve factory via isolated generator function</div><div>    prime_sieve_factory = build_prime_sieve(limit)</div> <div>    # 3. Test Forward Generation</div><div>    print("👉 Running Forward Traversal Validation...")</div><div>    actual_forward: List[int] = list(prime_sieve_factory(2, limit + 1, 1))</div><div>    </div><div>    if actual_forward == expected_forward:</div><div>        print(f"   ✅ PASS: Forward matching. Found {len(actual_forward)} primes.")</div><div>    else:</div><div>        print("   ❌ FAIL: Forward mismatch.")</div><div>        print(f"   Expected: {expected_forward}")</div><div>        print(f"   Got:      {actual_forward}")</div> <div>    # 4. Test Backward Generation</div><div>    print("👉 Running Backward Traversal Validation...")</div><div>    actual_backward: List[int] = list(prime_sieve_factory(limit, 1, -1))</div><div>    </div><div>    if actual_backward == expected_backward:</div><div>        print(f"   ✅ PASS: Backward matching. Reversed sequences match perfectly.")</div><div>    else:</div><div>        print("   ❌ FAIL: Backward mismatch.")</div><div>        print(f"   Expected: {expected_backward}")</div><div>        print(f"   Got:      {actual_backward}")</div> <div>if __name__ == "__main__":</div><div>    # Test with a limit of 200 elements to keep visualization digestible</div><div>    run_sieve_tests(limit=200)</div></div></div><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-247" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> </div></div></div><h4 data-path-to-node="28" style="font-family: Google Sans, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">The Verdict</h4>When running forward, the generator's internal pointers march sequentially alongside the main consumer loop.The true architectural challenge happens during the backward traversal pass (<code data-index-in-node="77" data-path-to-node="30" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">step = -1</code>). When executing <code data-index-in-node="104" data-path-to-node="30" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">prime_sieve_factory(limit, 1, -1)</code>, the engine must manage 13 independent composite exclusion ranges simultaneously (for m = 2, 3, ..., 14). Each sub-range pointer must instantly compute its upper boundary, snap precisely to its maximum valid multiple less than or equal to 200, and step backward in perfect synchronization without dropping or skipping values.The execution checks out perfectly:<div _ngcontent-ng-c2702060661="" class="code-block ng-tns-c2702060661-248 ng-animate-disabled ng-trigger ng-trigger-codeBlockRevealAnimation" data-hveid="0" data-ved="0CAAQhtANahgKEwi4k8qG4aaVAxUAAAAAHQAAAAAQgQU" style="display: block; font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="formatted-code-block-internal-container ng-tns-c2702060661-248" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="animated-opacity ng-tns-c2702060661-248" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><div _ngcontent-ng-c2702060661="" class="code-block-decoration header-formatted gds-emphasized-body-m ng-tns-c2702060661-248 ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"> <div _ngcontent-ng-c2702060661="" class="buttons ng-tns-c2702060661-248 ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><button _ngcontent-ng-c442952589="" aria-label="Download code" class="mdc-icon-button mat-mdc-icon-button mat-mdc-button-base mat-badge mat-unthemed mat-badge-overlap mat-badge-above mat-badge-after mat-badge-small mat-badge-hidden ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"></button><button _ngcontent-ng-c442952589="" aria-label="Copy code" class="mdc-icon-button mat-mdc-icon-button mat-mdc-button-base mat-badge mat-unthemed mat-badge-overlap mat-badge-above mat-badge-after mat-badge-small mat-badge-hidden ng-star-inserted" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"></button></div></div><pre _ngcontent-ng-c2702060661="" class="ng-tns-c2702060661-248" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;"><code _ngcontent-ng-c2702060661="" class="code-container formatted ng-tns-c2702060661-248" data-test-id="code-content" role="text" style="font-family: Google Sans Text, sans-serif !important; line-height: 1.15 !important; margin-top: 0px !important;">Beginning Sparse Range Sieve Evaluation Suite (N = 200)... ---------------------------------------------------------------- 👉 Running Forward Traversal Validation... ✅ PASS: Forward matching. Found 46 primes. 👉 Running Backward Traversal Validation... ✅ PASS: Backward matching. Reversed sequences match perfectly. </code></pre></div></div></div>If you are following the Python Discuss thread, this demonstrates that handling arbitrary sequence exclusions cleanly and statelessly via pure index-pointer stream math isn't just an elegant idea—it's highly reliable even under heavy algorithmic stress. DisclosureI used Gemini AI to help in this. I was able to state algorithms and get Gemini to fill in with implementations, state changes and get Gemini to implement those too. I am used to creating algorithms without AI, this time I was able to state what I wanted in some detail and have Gemini add yet more detail. I could show python, pseudo-code, r textual descriptions as needed and get Gemini to fill in. The tests went from my description to code by Gemini - the sieve example test idea and spec, and debug was by me with Gemini improving my limit on m from m*2 to m*m for example before implementing as told.</div>Sat, 27 Jun 2026 16:19:06 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