https://cs.lozic.me/Recent content on CS TheoremsHugoen-usFri, 06 Aug 2027 12:00:00 +0100https://cs.lozic.me/posts/t040-the-myhill-nerode-theorem/Fri, 06 Aug 2027 12:00:00 +0100https://cs.lozic.me/posts/t040-the-myhill-nerode-theorem/<h2 id="symptom">Symptom</h2> <p>You need to prove a language is not regular. Everyone points you at the pumping lemma, and you spend an afternoon losing to it.</p> <p>The statement is a nest of quantifiers: for every regular language there exists a pumping length $p$ such that for every string $w$ with $|w| \ge p$ there exists a decomposition $w = xyz$ with $|xy| \le p$ and $|y| &gt; 0$ such that for all $i \ge 0$, $xy^i z$ is in the language. To use it you negate all of that and play a game against an adversary who picks $p$ and the decomposition while you pick $w$ and $i$.</p>https://cs.lozic.me/posts/t039-kleenes-theorem-regexes-are-finite-automata/Fri, 30 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t039-kleenes-theorem-regexes-are-finite-automata/<h2 id="symptom">Symptom</h2> <p>Your service goes down. CPU pinned at 100% on one core, no memory growth, no error logs, requests timing out.</p> <p>The cause is a validation regex — something reasonable-looking like <code>^(a+)+$</code>, or in real life a URL or email validator with nested quantifiers — matched against a 30-character string that happens not to match. The engine is exploring an exponential number of ways to split the input among the nested groups, and it will get there eventually, some time after the heat death of everything.</p>https://cs.lozic.me/posts/t035-bpp-pseudorandomness-and-derandomization/Fri, 23 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t035-bpp-pseudorandomness-and-derandomization/<h2 id="symptom">Symptom</h2> <p>You need to test whether two arithmetic circuits compute the same polynomial. The deterministic approach is to expand both into normal form, and the expansion is exponentially large.</p> <p>The randomized approach takes four lines. Pick a random point, evaluate both circuits there, compare. If they differ as polynomials, the Schwartz-Zippel lemma says a random point catches it with probability at least $1 - d/|S|$. Repeat a few times and the error is negligible.</p>https://cs.lozic.me/posts/t037-the-exponential-time-hypothesis-and-fine-grained-complexity/Fri, 16 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t037-the-exponential-time-hypothesis-and-fine-grained-complexity/<h2 id="symptom">Symptom</h2> <p>You have a string algorithm. Edit distance between two sequences, the classic dynamic program, $O(n^2)$ time. It has been in production for years.</p> <p>Now the inputs are genome-scale. At $n = 10^5$ characters, $n^2 = 10^{10}$ operations, about <strong>10 seconds</strong> at a billion ops per second. At $n = 10^6$ it is $10^{12}$ operations, about <strong>1000 seconds</strong> — seventeen minutes for one pair of strings. You need to do a million pairs.</p>https://cs.lozic.me/posts/t034-parity-is-not-in-ac0/Fri, 09 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t034-parity-is-not-in-ac0/<h2 id="symptom">Symptom</h2> <p>You have just read T030 and you are demoralized. Relativization kills the techniques that treat machines as black boxes. Natural proofs kill the combinatorial ones, assuming pseudorandom functions exist. Between them they appear to rule out everything anyone knows how to do, and the honest summary of fifty years is that nobody can prove any interesting problem is hard.</p>https://cs.lozic.me/posts/t110-the-no-cloning-theorem/Fri, 02 Jul 2027 12:00:00 +0100https://cs.lozic.me/posts/t110-the-no-cloning-theorem/<h2 id="symptom">Symptom</h2> <p>You have spent this whole part learning that information wants to be copied. Entropy bounds compression (T018), Huffman hits the bound (T020), redundancy buys error correction (T021). Every result so far treats &ldquo;make another copy&rdquo; as the free operation — the thing you do without thinking, the reason <code>cp</code> has no interesting failure modes.</p>https://cs.lozic.me/posts/t106-the-nyquist-shannon-sampling-theorem/Fri, 25 Jun 2027 12:00:00 +0100https://cs.lozic.me/posts/t106-the-nyquist-shannon-sampling-theorem/<h2 id="symptom">Symptom</h2> <p>Your dashboard shows CPU utilization averaging 40%, comfortably under budget. Users are reporting timeouts. You add more logging and the average stays at 40%.</p> <p>Eventually someone captures a one-second trace and finds the truth: the service spikes to 100% for eight seconds out of every ninety, and your monitoring samples every sixty seconds. The spikes are real, periodic, and invisible.</p>https://cs.lozic.me/posts/t063-the-byzantine-generals-problem/Fri, 18 Jun 2027 12:00:00 +0100https://cs.lozic.me/posts/t063-the-byzantine-generals-problem/<h2 id="symptom">Symptom</h2> <p>A node in your cluster is not down. It is worse than down.</p> <p>Its disk is returning corrupted blocks that pass the checksum because the checksum is corrupted too. Or a NIC with a firmware bug is duplicating and reordering frames. Or a bad memory module flipped a bit in a config value and the node now believes it owns a shard it does not. Or someone has root on one machine.</p>https://cs.lozic.me/posts/t062-flp-impossibility/Fri, 11 Jun 2027 12:00:00 +0100https://cs.lozic.me/posts/t062-flp-impossibility/<h2 id="symptom">Symptom</h2> <p>Your cluster of five nodes elects a leader. It works. It has worked for a year.</p> <p>Then one afternoon a garbage collection pause on the leader runs for twelve seconds. The followers time out, start an election, and elect a new leader. The old leader wakes up, has no idea it was deposed, and keeps serving writes. For a few hundred milliseconds you have two leaders, and if your fencing is not airtight, two writes that should have been ordered land in an order nobody chose.</p>https://cs.lozic.me/posts/t059-vector-clocks-and-causal-consistency/Fri, 04 Jun 2027 12:00:00 +0100https://cs.lozic.me/posts/t059-vector-clocks-and-causal-consistency/<h2 id="symptom">Symptom</h2> <p>You took the lesson from T058 and stopped trusting wall clocks. Every event now carries a Lamport timestamp, and your writes are ordered by it. The vanishing updates stopped.</p>https://cs.lozic.me/posts/t058-happens-before-and-lamport-clocks/Fri, 28 May 2027 12:00:00 +0100https://cs.lozic.me/posts/t058-happens-before-and-lamport-clocks/<h2 id="symptom">Symptom</h2> <p>Two writes to the same key. Which one wins?</p> <p>Your first instinct is timestamps: whichever has the later wall-clock time is newer. This works in testing. In production you get a bug report saying an update vanished, and when you dig in, node B&rsquo;s clock was 40 ms behind node A&rsquo;s, so B&rsquo;s <em>later</em> write carried an <em>earlier</em> timestamp and lost. NTP is running. NTP does not make clocks identical, it makes them close, and &ldquo;close&rdquo; is not &ldquo;ordered.&rdquo;</p>https://cs.lozic.me/posts/t033-parameterized-complexity-fpt-and-w1/Fri, 21 May 2027 12:00:00 +0100https://cs.lozic.me/posts/t033-parameterized-complexity-fpt-and-w1/<h2 id="symptom">Symptom</h2> <p>Your problem is NP-hard, and yet it keeps being easy.</p> <p>You are computing a minimum vertex cover to select monitoring points in a network. Twelve thousand nodes. NP-hard, per T028, so you brace for the worst — and the answer comes back in under a second, every time. The cover is small, around thirty nodes, because the network is sparse and the interesting nodes are few.</p>https://cs.lozic.me/posts/t032-approximation-algorithms-and-ratios/Fri, 14 May 2027 12:00:00 +0100https://cs.lozic.me/posts/t032-approximation-algorithms-and-ratios/<h2 id="symptom">Symptom</h2> <p>You have accepted that the problem is NP-hard. Now what?</p> <p>The literature offers a wall of results with numbers attached: 2-approximation, $\ln n$-approximation, PTAS, FPTAS, 0.878. Nobody explains what these numbers buy you, how they are proved, or how they connect to the heuristic you already wrote. Meanwhile your greedy heuristic is running in production and you have no idea whether it is within 5% or a factor of 50 of optimal, because <strong>you cannot compute the optimum to compare against</strong> — that was the whole problem.</p>https://cs.lozic.me/posts/t031-the-pcp-theorem-and-inapproximability/Fri, 07 May 2027 12:00:00 +0100https://cs.lozic.me/posts/t031-the-pcp-theorem-and-inapproximability/<h2 id="symptom">Symptom</h2> <p>You gave up on exact (T028) and went looking for an approximation.</p> <p>Vertex cover: you find a 2-approximation in four lines. Knapsack: an FPTAS, any accuracy you want. Encouraged, you go looking for the same for max-clique, and you find nothing. Not &ldquo;nothing good&rdquo; — the best known ratio is around $n/(\log n)^2$, which on a 10,000-vertex graph means the algorithm might return a clique nearly 60 times smaller than the true maximum. That is not an approximation, it is a rumour.</p>https://cs.lozic.me/posts/t105-streaming-lower-bounds-and-sketching/Fri, 30 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t105-streaming-lower-bounds-and-sketching/<h2 id="symptom">Symptom</h2> <p>Product wants the daily unique-visitor count. You have a firehose of events.</p> <p>The obvious implementation is a set. Add each visitor ID, report the size. At a billion distinct IDs, eight bytes each, that is 8 GB before any hash table overhead, and in practice a <code>HashSet</code> will cost you two to three times that. Per day. Per dimension you want to slice by. Multiply by country, by platform, by campaign, and the memory bill is absurd for a number nobody looks at past two significant figures.</p>https://cs.lozic.me/posts/t104-universal-hashing/Fri, 23 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t104-universal-hashing/<h2 id="symptom">Symptom</h2> <p>Your service went down under a hash collision attack.</p> <p>Somebody noticed your web framework put POST parameters into a hash table, found thousands of distinct keys colliding under its hash function, and posted a form with 20,000 of them. Every insert walked a chain. Quadratic behaviour, one CPU pinned per request, service dead. This actually happened, across PHP, Python, Ruby, Java and .NET in 2011, and again against Rust&rsquo;s default <code>HashMap</code> before it switched to SipHash.</p>https://cs.lozic.me/posts/t025-minimum-description-length-and-occams-razor/Fri, 16 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t025-minimum-description-length-and-occams-razor/<h2 id="symptom">Symptom</h2> <p>You fit a model. It scores 94% on training data and 71% on held-out data.</p> <p>So you simplify: fewer parameters, more regularization. Training drops to 88%, held-out rises to 84%. You simplify further and both drop. Somewhere in there was an optimum, and you found it by trial and error, with a validation set and patience.</p>https://cs.lozic.me/posts/t023-kolmogorov-complexity/Fri, 09 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t023-kolmogorov-complexity/<h2 id="symptom">Symptom</h2> <p>You have two files, each exactly one megabyte.</p> <p>The first is a megabyte of the byte <code>0x00</code>. The second is a megabyte from <code>/dev/urandom</code>. Gzip the first and you get a few hundred bytes. Gzip the second and you get slightly more than a megabyte, because the header costs something and there is nothing to exploit.</p>https://cs.lozic.me/posts/t030-the-barriers-relativization-and-natural-proofs/Fri, 02 Apr 2027 12:00:00 +0100https://cs.lozic.me/posts/t030-the-barriers-relativization-and-natural-proofs/<h2 id="symptom">Symptom</h2> <p>You have read T026 and T027 and you are wondering the obvious thing.</p> <p>Fifty years. Thousands of researchers. A million-dollar prize. And P vs NP has not moved. That is strange — most famous problems yield partial results, near misses, special cases that suggest the shape of the answer. Here there is almost nothing. We cannot even prove that NP requires more than linear-size circuits, which is a laughably weak statement compared to what everyone believes.</p>https://cs.lozic.me/posts/t028-karps-21-problems/Fri, 26 Mar 2027 12:00:00 +0100https://cs.lozic.me/posts/t028-karps-21-problems/<h2 id="symptom">Symptom</h2> <p>Three weeks into a project, you are still trying to write an exact algorithm.</p> <p>The problem is yours and it looks specific: assign delivery vans to routes such that every stop is covered, no van exceeds its capacity, and the total distance is minimized. Nothing in the literature matches it exactly. So you keep going — better data structures, smarter pruning, a cleverer greedy pass with a repair step — and each version works on your test set and falls over on production data.</p>https://cs.lozic.me/posts/t027-the-cook-levin-theorem/Fri, 19 Mar 2027 12:00:00 +0100https://cs.lozic.me/posts/t027-the-cook-levin-theorem/<h2 id="symptom">Symptom</h2> <p>You know reductions (T017): to prove your problem hard, reduce a known-hard problem to it. Fine. But that begs the obvious question, and it is the question a sharp colleague asks the first time you use the technique.</p>https://cs.lozic.me/posts/t026-p-vs-np/Fri, 12 Mar 2027 12:00:00 +0100https://cs.lozic.me/posts/t026-p-vs-np/<h2 id="symptom">Symptom</h2> <p>You are handed a scheduling problem. Two hundred tasks, precedence constraints, shared resources, and a deadline. Somebody wants an optimal schedule.</p> <p>Checking a proposed schedule takes seconds: walk the list, verify each constraint, add up the makespan. Finding one is different. Your search runs for a day and finds nothing. You try simulated annealing, then a genetic algorithm, then branch and bound. Each gives good schedules and none gives the best one, and you cannot tell whether the problem is hard or you are not clever enough.</p>https://cs.lozic.me/posts/t022-the-hamming-bound/Fri, 05 Mar 2027 12:00:00 +0100https://cs.lozic.me/posts/t022-the-hamming-bound/<h2 id="symptom">Symptom</h2> <p>You are specifying ECC memory for a fleet. The vendor quotes SECDED: single error correct, double error detect, at 8 check bits per 64 data bits. That is 12.5% overhead, and someone in the room asks the obvious question — why not correct two errors? Or three? What does it cost?</p>https://cs.lozic.me/posts/t021-the-noisy-channel-coding-theorem/Fri, 26 Feb 2027 12:00:00 +0100https://cs.lozic.me/posts/t021-the-noisy-channel-coding-theorem/<h2 id="symptom">Symptom</h2> <p>Your link drops 1% of bits. You need reliable delivery, so you triple every bit and take a majority vote. Now an error needs two of three bits to flip, which happens with probability about $3 \times 10^{-4}$ — better, but you paid 3x in bandwidth and the error rate is still not zero.</p>https://cs.lozic.me/posts/t020-huffman-coding-is-optimal/Fri, 19 Feb 2027 12:00:00 +0100https://cs.lozic.me/posts/t020-huffman-coding-is-optimal/<h2 id="symptom">Symptom</h2> <p>You are compressing a log file. The symbols are wildly skewed: <code>INFO</code> is 90% of the lines, <code>WARN</code> is 9%, <code>ERROR</code> and <code>FATAL</code> split the rest. Fixed-width two-bit codes give you exactly 2 bits per symbol, and entropy (T018) says the floor is about 0.53 bits. There is a factor of four sitting there.</p>https://cs.lozic.me/posts/t009-balls-into-bins/Fri, 12 Feb 2027 12:00:00 +0100https://cs.lozic.me/posts/t009-balls-into-bins/<h2 id="symptom">Symptom</h2> <p>You have 100 backends and a load balancer hashing request IDs to pick one. Expected load per backend is exactly 1%, and you have checked the hash is good.</p>https://cs.lozic.me/posts/t017-reductions/Fri, 05 Feb 2027 12:00:00 +0100https://cs.lozic.me/posts/t017-reductions/<h2 id="symptom">Symptom</h2> <p>You have a new problem. Your build system needs to decide whether two configuration files can ever produce conflicting outputs. Or your linter needs to decide whether a regex with backreferences can match a string of a given length. Or your scheduler needs to decide whether a set of periodic tasks is feasible.</p>https://cs.lozic.me/posts/t051-the-coffman-conditions/Fri, 29 Jan 2027 12:00:00 +0100https://cs.lozic.me/posts/t051-the-coffman-conditions/<h2 id="symptom">Symptom</h2> <p>Production is wedged. Not slow — wedged. CPU is at zero, no errors are being logged, and two threads are sitting in a lock wait with no timeout. The stack traces show thread A holding the account lock and waiting for the ledger lock, thread B holding the ledger lock and waiting for the account lock. Somebody restarts the service and it goes away.</p>https://cs.lozic.me/posts/t015-godels-incompleteness-theorems/Fri, 22 Jan 2027 12:00:00 +0100https://cs.lozic.me/posts/t015-godels-incompleteness-theorems/<h2 id="symptom">Symptom</h2> <p>The proof assistant is failing you. You are trying to verify a piece of concurrent code in Coq or Lean, and the tool will not accept your termination argument. A colleague says, without much conviction, &ldquo;well, Gödel — you can&rsquo;t prove everything anyway.&rdquo;</p>https://cs.lozic.me/posts/t014-kleenes-recursion-theorem/Fri, 15 Jan 2027 12:00:00 +0100https://cs.lozic.me/posts/t014-kleenes-recursion-theorem/<h2 id="symptom">Symptom</h2> <p>Write a program that prints its own source code. No file I/O, no reading <code>__file__</code>, no cheating.</p> <p>The first attempt fails instantly. To print the source you must contain the source, and then the containing text is also part of the source, so you must contain that too. The regress is obviously infinite, and most people conclude after ten minutes that the task is impossible.</p>https://cs.lozic.me/posts/t038-the-chomsky-hierarchy/Fri, 08 Jan 2027 12:00:00 +0100https://cs.lozic.me/posts/t038-the-chomsky-hierarchy/<h2 id="symptom">Symptom</h2> <p>The most upvoted answer in Stack Overflow history is a refusal to answer. Someone asked how to match nested HTML tags with a regular expression, and the reply is a page of escalating horror about the centre not holding. It is funny, and it is also a theorem, which is not obvious from reading it.</p>https://cs.lozic.me/posts/t013-the-universal-machine/Fri, 01 Jan 2027 12:00:00 +0100https://cs.lozic.me/posts/t013-the-universal-machine/<h2 id="symptom">Symptom</h2> <p>You write a Python program. It runs. What ran it?</p> <p>CPython — itself a program, written in C, compiled to x86 instructions, executed by a CPU whose control unit is arguably interpreting microcode, possibly inside a virtual machine, on a kernel that scheduled it, all perhaps within a container image. At no point in that stack does anyone find it strange that a program&rsquo;s job is running other programs. We build emulators, JITs, WebAssembly runtimes, Docker, QEMU, and browsers that download and execute arbitrary code from strangers, and treat every layer as ordinary engineering.</p>https://cs.lozic.me/posts/t012-the-church-turing-thesis/Fri, 25 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t012-the-church-turing-thesis/<h2 id="symptom">Symptom</h2> <p>Somebody says &ldquo;CSS is Turing-complete&rdquo; and the room splits. Half the people treat it as a joke about a styling language, the other half as a serious claim with consequences. Both reactions are common and only one is right.</p>https://cs.lozic.me/posts/t008-concentration-bounds/Fri, 18 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t008-concentration-bounds/<h2 id="symptom">Symptom</h2> <p>Your service handles four billion requests a month and you want the error rate. Computing it exactly means a job over four billion log lines. Someone suggests sampling ten thousand of them.</p>https://cs.lozic.me/posts/t006-amortized-analysis-and-the-potential-method/Fri, 11 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t006-amortized-analysis-and-the-potential-method/<h2 id="symptom">Symptom</h2> <p>You have a dynamic array. <code>push</code> writes one element and bumps a counter, which is clearly $O(1)$ — except when the array is full, in which case it allocates a new buffer of twice the size, copies every element across, and frees the old one. That is $O(n)$.</p>https://cs.lozic.me/posts/t005-adversary-arguments/Fri, 04 Dec 2026 12:00:00 +0100https://cs.lozic.me/posts/t005-adversary-arguments/<h2 id="symptom">Symptom</h2> <p>You need the largest and second-largest element of an array. The obvious way is two passes: $n-1$ comparisons for the max, then $n-2$ for the max of the rest. That is $2n - 3$.</p>https://cs.lozic.me/posts/t007-linearity-of-expectation/Fri, 27 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t007-linearity-of-expectation/<h2 id="symptom">Symptom</h2> <p>You are analyzing a hash table. You want the expected number of buckets that end up empty. The bucket occupancies are all tangled together — one key landing in bucket 3 makes every other bucket slightly less likely to be chosen — and the dependencies look like they will make the sum intractable.</p>https://cs.lozic.me/posts/t003-comparison-sort-lower-bound/Fri, 20 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t003-comparison-sort-lower-bound/<h2 id="symptom">Symptom</h2> <p>Every general-purpose sort in every standard library is $O(n \log n)$. Timsort, introsort, pdqsort, merge sort, heapsort. Decades of work by extremely motivated people, an enormous amount of money riding on it, and they all land on the same exponent.</p>https://cs.lozic.me/posts/t018-shannon-entropy-and-source-coding/Fri, 13 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t018-shannon-entropy-and-source-coding/<h2 id="symptom">Symptom</h2> <p>You gzip a log file and it drops to 8% of its original size. You gzip the already-gzipped file and it gets slightly <em>bigger</em>. Somebody asks why, and the honest answer you have is &ldquo;because it&rsquo;s already compressed,&rdquo; which is a restatement, not a reason.</p>https://cs.lozic.me/posts/t074-the-birthday-bound/Fri, 06 Nov 2026 12:00:00 +0100https://cs.lozic.me/posts/t074-the-birthday-bound/<h2 id="symptom">Symptom</h2> <p>You need a short ID for uploads. Eight hex characters feels generous, so you take the first 32 bits of a hash and move on.</p> <p>At about 80,000 uploads, two files collide, and one of them silently overwrites the other, and the bug report says the customer&rsquo;s invoice contains someone else&rsquo;s line items.</p>https://cs.lozic.me/posts/t061-the-two-generals-problem/Fri, 30 Oct 2026 12:00:00 +0100https://cs.lozic.me/posts/t061-the-two-generals-problem/<h2 id="symptom">Symptom</h2> <p>You call the payment API. The request goes out. Nothing comes back.</p> <p>Did the charge happen? You genuinely do not know. Retrying might double-charge the customer. Not retrying might drop the payment. There is no third option available, and no amount of care in your client library creates one.</p>https://cs.lozic.me/posts/t089-conways-law/Fri, 23 Oct 2026 12:00:00 +0200https://cs.lozic.me/posts/t089-conways-law/<h2 id="symptom">Symptom</h2> <p>The service has four layers. Nobody designed four layers. There were four teams.</p> <p>Or the version that arrives in a postmortem: the outage happened at the boundary between two services, in the retry logic, because each side assumed the other owned the deduplication. Both teams were right about their own component and neither had ever been in a room with the other.</p>https://cs.lozic.me/posts/t011-rices-theorem/Fri, 16 Oct 2026 12:00:00 +0200https://cs.lozic.me/posts/t011-rices-theorem/<h2 id="symptom">Symptom</h2> <p>The security team asks for a scanner with no false positives and no false negatives. Every piece of malware caught, nothing legitimate quarantined.</p> <p>The platform team asks whether the analyzer can flag every function that performs I/O, so the pure ones can be cached automatically.</p>https://cs.lozic.me/posts/t010-the-halting-problem/Fri, 09 Oct 2026 12:00:00 +0200https://cs.lozic.me/posts/t010-the-halting-problem/<h2 id="symptom">Symptom</h2> <p>Your CI has a test that hangs. Not fails — hangs. Somebody suggests the obvious fix: before running a test, check whether it terminates, and skip it if not.</p>https://cs.lozic.me/posts/t002-cantors-diagonal-argument/Fri, 02 Oct 2026 12:00:00 +0200https://cs.lozic.me/posts/t002-cantors-diagonal-argument/<h2 id="symptom">Symptom</h2> <p>Somebody proposes a tool that will find all the bugs. Not most bugs. All of them. The pitch has the shape: we enumerate the failure modes, we write a checker for each, we keep adding checkers, and the set of undetected bugs shrinks toward zero.</p>https://cs.lozic.me/posts/t082-gustafsons-law-and-the-usl/Fri, 25 Sep 2026 12:00:00 +0200https://cs.lozic.me/posts/t082-gustafsons-law-and-the-usl/<h2 id="symptom">Symptom</h2> <p>Two measurements that both look like they break the previous post.</p> <p>The first: your Spark job takes four hours on ten nodes. You move it to a hundred nodes and it still takes four hours — but it is now processing ten times the data. <a href="https://cs.lozic.me/posts/t081-amdahls-law/">T081</a> said the ceiling was 20×, and nobody hit a ceiling.</p>https://cs.lozic.me/posts/t081-amdahls-law/Fri, 18 Sep 2026 12:00:00 +0200https://cs.lozic.me/posts/t081-amdahls-law/<h2 id="symptom">Symptom</h2> <p>The profile was unambiguous: 94% of wall-clock time in one loop, and the loop&rsquo;s iterations are independent. You parallelise it across sixteen cores, the flame graph flattens exactly as predicted, and the end-to-end time goes from 40 seconds to 17.</p>https://cs.lozic.me/posts/t091-hyrums-law-and-postels-principle/Fri, 11 Sep 2026 12:00:00 +0200https://cs.lozic.me/posts/t091-hyrums-law-and-postels-principle/<h2 id="symptom">Symptom</h2> <p>You fix a typo in an error message. Four days later a customer escalates: their alerting pipeline greps for the old string and has gone quiet.</p> <p>Or you make a function faster, and a test suite that had been green for two years starts failing intermittently — not because the function is wrong, but because it used to be slow enough to hide a race.</p>https://cs.lozic.me/posts/t004-no-universal-lossless-compressor/Fri, 04 Sep 2026 12:00:00 +0200https://cs.lozic.me/posts/t004-no-universal-lossless-compressor/<h2 id="symptom">Symptom</h2> <p>Someone sends you a pitch deck. The claim is a compression algorithm that reduces <em>any</em> file by a guaranteed ratio, and — this is always the tell — that it can be applied repeatedly. Four gigabytes down to one, then one down to two fifty, and so on until the whole film is a few hundred bytes and a clever decoder.</p>https://cs.lozic.me/posts/t001-pigeonhole-principle/Fri, 28 Aug 2026 12:00:00 +0200https://cs.lozic.me/posts/t001-pigeonhole-principle/<h2 id="symptom">Symptom</h2> <p>You are asked to build a cache key. You have a 64-bit hash, and someone on the team says: &ldquo;collisions are basically impossible, there are eighteen quintillion values.&rdquo;</p> <p>Or: your service assigns short IDs to uploads, six characters of base-36, and you are wondering when you need to worry.</p>https://cs.lozic.me/index-of-results/Mon, 01 Jan 0001 00:00:00 +0000https://cs.lozic.me/index-of-results/Every result in the series, in book order.