InkdownInkdown
Start writing

Arpit Bhayani Blogs

336 files·168 subfolders

Shared Workspace

Arpit Bhayani Blogs
001 Ai Topological Sort

159-rule-30-cellular-automata

Shared from "Arpit Bhayani Blogs" on Inkdown

Rule 30 - Generating Random Numbers with Cellular Automata

Source: https://arpitbhayani.me/blogs/rule-30-cellular-automata Date: 2020-02-14

Explore Rule 30, a cellular automaton, and its surprising application in pseudorandom number generation. Discover its chaotic patterns!

A pseudorandom number generator produces numbers deterministically but they seem aperiodic (random) most of the time for most use-cases. The generator accepts a seed value (ideally a true random number) and starts producing the sequence as a function of this seed and/or a previous number of the sequence. These are Pseudorandom (not truly random) because if seed value is known they can be determined algorithmically. True random numbers are hardware generated or generated from blood volume pulse, atmospheric pressure, thermal noise, quantum phenomenon, etc.

There are lots of techniques to generate Pseudorandom numbers, namely: , , , etc. Today we dive deep into that uses a controversial science called . This method passes many standard tests for randomness and was used in for generating random integers.

001-ai-topological-sort.md
tldr.md
002 Temporal Primer
002-temporal-primer.md
tldr.md
003 Rag Production
003-rag-production.md
tldr.md
004 Structure Of Llm Chat
004-structure-of-llm-chat.md
tldr.md
005 How Llms Work
005-how-llms-work.md
tldr.md
006 Monolith Is Distributed System
006-monolith-is-distributed-system.md
tldr.md
007 Defensive Databases
007-defensive-databases.md
tldr.md
008 Bm25
008-bm25.md
tldr.md
009 Join Algorithms
009-join-algorithms.md
tldr.md
010 Venting At Work
010-venting-at-work.md
tldr.md
011 Half Life
011-half-life.md
tldr.md
012 Multi Paxos
012-multi-paxos.md
tldr.md
013 Mysql Replication Internals
013-mysql-replication-internals.md
tldr.md
014 Bloom Filters
014-bloom-filters.md
tldr.md
015 Clock Sync Nightmare
015-clock-sync-nightmare.md
tldr.md
016 Kafka Partitions
016-kafka-partitions.md
tldr.md
017 Product Quantization
017-product-quantization.md
tldr.md
018 Qkv Matrices
018-qkv-matrices.md
tldr.md
019 Deleted Production
019-deleted-production.md
tldr.md
020 How Llm Inference Works
020-how-llm-inference-works.md
tldr.md
021 Blocking Queues
021-blocking-queues.md
tldr.md
022 Heartbeats In Distributed Systems
022-heartbeats-in-distributed-systems.md
tldr.md
023 Cassandra Writes
023-cassandra-writes.md
tldr.md
024 Redis Replication
024-redis-replication.md
tldr.md
025 Arrogant People At Work
025-arrogant-people-at-work.md
tldr.md
026 Cdn Content Replication
026-cdn-content-replication.md
tldr.md
027 Cant Fix Everything Day One
027-cant-fix-everything-day-one.md
tldr.md
028 Emotions At Work
028-emotions-at-work.md
tldr.md
029 Grpc Http2
029-grpc-http2.md
tldr.md
030 Meetings With No Agenda Are A Waste Of Time
030-meetings-with-no-agenda-are-a-waste-of-time.md
tldr.md
031 Growth Is Not About Doing Everything
031-growth-is-not-about-doing-everything.md
tldr.md
032 Career Longevity Vs Job Hopping
032-career-longevity-vs-job-hopping.md
tldr.md
033 Stay Relevant At Higher Salary Levels
033-stay-relevant-at-higher-salary-levels.md
tldr.md
034 Why Consensus
034-why-consensus.md
tldr.md
035 Database Deadlocks
035-database-deadlocks.md
tldr.md
036 Cpu Cache Locality
036-cpu-cache-locality.md
tldr.md
037 Eventual Consistency
037-eventual-consistency.md
tldr.md
038 Dns Udp Tcp
038-dns-udp-tcp.md
tldr.md
039 Masters
039-masters.md
tldr.md
040 Empathy Makes Great Engineers Unstoppable
040-empathy-makes-great-engineers-unstoppable.md
tldr.md
041 Good Mentors Build People
041-good-mentors-build-people.md
tldr.md
042 Always Have Back Burner Projects
042-always-have-back-burner-projects.md
tldr.md
043 Before You Push Back Know What Youre Standing On
043-before-you-push-back-know-what-youre-standing-on.md
tldr.md
044 Be The One They Can Count On
044-be-the-one-they-can-count-on.md
tldr.md
045 How Much People Bet On You
045-how-much-people-bet-on-you.md
tldr.md
046 How To Get Leadership To Say Yes To Your Project
046-how-to-get-leadership-to-say-yes-to-your-project.md
tldr.md
047 Dont Let Your Best Ideas Die In Silence
047-dont-let-your-best-ideas-die-in-silence.md
tldr.md
048 Be Someone Others Want To Work With
048-be-someone-others-want-to-work-with.md
tldr.md
049 Dont Fall For Xy Problem Ask Right Questions
049-dont-fall-for-xy-problem-ask-right-questions.md
tldr.md
050 Biggest Lie Startups Tell Engineers
050-biggest-lie-startups-tell-engineers.md
tldr.md
051 Promotions Are Proactive Not Reactive
051-promotions-are-proactive-not-reactive.md
tldr.md
052 Not Enough To Be Right Learn To Be Heard
052-not-enough-to-be-right-learn-to-be-heard.md
tldr.md
053 No One Ships Alone
053-no-one-ships-alone.md
tldr.md
054 Not Every Mistake Needs A Correction
054-not-every-mistake-needs-a-correction.md
tldr.md
055 Build Influence At Work
055-build-influence-at-work.md
tldr.md
056 Your Soft Skills Arent Soft At All
056-your-soft-skills-arent-soft-at-all.md
tldr.md
057 Experience Before Forming Opinion
057-experience-before-forming-opinion.md
tldr.md
058 Curiosity And High Bias For Action
058-curiosity-and-high-bias-for-action.md
tldr.md
059 Worklog
059-worklog.md
tldr.md
060 Mistakes And Growth
060-mistakes-and-growth.md
tldr.md
061 Own It Instead Of Sweeping It Aside
061-own-it-instead-of-sweeping-it-aside.md
tldr.md
062 Dont Wait Step Up
062-dont-wait-step-up.md
tldr.md
063 Temporary Fix Is Permanent
063-temporary-fix-is-permanent.md
tldr.md
064 Interview Bias And What Sets You Apart
064-interview-bias-and-what-sets-you-apart.md
tldr.md
065 Saying This Isnt My Problem Is A Problem
065-saying-this-isnt-my-problem-is-a-problem.md
tldr.md
066 Okr
066-okr.md
tldr.md
067 Miscommunication
067-miscommunication.md
tldr.md
068 When In Doubt Code It Out
068-when-in-doubt-code-it-out.md
tldr.md
069 Follow Up Without Annoying People
069-follow-up-without-annoying-people.md
tldr.md
070 Lead Projects That Land
070-lead-projects-that-land.md
tldr.md
071 Abstract Thinking Skill Next Decade
071-abstract-thinking-skill-next-decade.md
tldr.md
072 We Engineers Suck At Task Estimation
072-we-engineers-suck-at-task-estimation.md
tldr.md
073 Shiny Object Syndrome In Tech
073-shiny-object-syndrome-in-tech.md
tldr.md
074 3p
074-3p.md
tldr.md
075 Leverage The Equilibrium
075-leverage-the-equilibrium.md
tldr.md
076 On Demand Container Loading In Aws Lambda
076-on-demand-container-loading-in-aws-lambda.md
tldr.md
077 Sql Has Problems We Can Fix Them Pipe Syntax In Sql
077-sql-has-problems-we-can-fix-them-pipe-syntax-in-sql.md
tldr.md
078 Nanolog A Nanosecond Scale Logging System
078-nanolog-a-nanosecond-scale-logging-system.md
tldr.md
079 Best Resource Is Mythical
079-best-resource-is-mythical.md
tldr.md
080 Wtf The Who To Follow Service At Twitter
080-wtf-the-who-to-follow-service-at-twitter.md
tldr.md
081 Know A Lot
081-know-a-lot.md
tldr.md
082 Out Of Syllabus
082-out-of-syllabus.md
tldr.md
083 Negotiate The Offer
083-negotiate-the-offer.md
tldr.md
084 Never Bad Mouth Your Ex Exployer
084-never-bad-mouth-your-ex-exployer.md
tldr.md
085 Culture Fit
085-culture-fit.md
tldr.md
086 Quantification In Resume
086-quantification-in-resume.md
tldr.md
087 Hiring Is Unfair
087-hiring-is-unfair.md
tldr.md
088 Questions For Interviewers
088-questions-for-interviewers.md
tldr.md
089 Collaboration Communication
089-collaboration-communication.md
tldr.md
090 Out Of Vicious Interview Cycle
090-out-of-vicious-interview-cycle.md
tldr.md
091 Pitch Projects Not Ideas
091-pitch-projects-not-ideas.md
tldr.md
092 Read Design Docs
092-read-design-docs.md
tldr.md
093 Read Rca Docs
093-read-rca-docs.md
tldr.md
094 Start Generalist
094-start-generalist.md
tldr.md
095 Do Not Rely On Summaries
095-do-not-rely-on-summaries.md
tldr.md
096 Structure Your Design Interviews
096-structure-your-design-interviews.md
tldr.md
097 Title Inflation
097-title-inflation.md
tldr.md
098 Find Your Own Project
098-find-your-own-project.md
tldr.md
099 Six Pointers To Crack Coding And Design Interviews
099-six-pointers-to-crack-coding-and-design-interviews.md
tldr.md
100 Keep Yourself Unblocked
100-keep-yourself-unblocked.md
tldr.md
101 Genetic Knapsack
101-genetic-knapsack.md
tldr.md
102 Pseudorandom Number Generation Lfsr
102-pseudorandom-number-generation-lfsr.md
tldr.md
103 How Indexes Work On Partitioned And Sharded Data
103-how-indexes-work-on-partitioned-and-sharded-data.md
tldr.md
104 Some Data Partitioning Strategies For Distributed Data Stores
104-some-data-partitioning-strategies-for-distributed-data-stores.md
tldr.md
105 Data Partitioning
105-data-partitioning.md
tldr.md
106 Leaderless Replication
106-leaderless-replication.md
tldr.md
107 Conflict Resolution
107-conflict-resolution.md
tldr.md
108 Conflict Detection
108-conflict-detection.md
tldr.md
109 Multi Master Replication
109-multi-master-replication.md
tldr.md
110 Monotonic Reads
110-monotonic-reads.md
tldr.md
111 Read Your Write Consistency
111-read-your-write-consistency.md
tldr.md
112 Handling Outages Master Replica
112-handling-outages-master-replica.md
tldr.md
113 Replication Formats
113-replication-formats.md
tldr.md
114 Replication Strategies
114-replication-strategies.md
tldr.md
115 Master Replica Replication
115-master-replica-replication.md
tldr.md
116 Durability
116-durability.md
tldr.md
117 Isolation
117-isolation.md
tldr.md
118 Atomicity
118-atomicity.md
tldr.md
119 Consistency
119-consistency.md
tldr.md
120 Architectures In Distributed Systems
120-architectures-in-distributed-systems.md
tldr.md
121 Mistaken Beliefs Of Distributed Systems
121-mistaken-beliefs-of-distributed-systems.md
tldr.md
122 Fork Bomb
122-fork-bomb.md
tldr.md
123 Chained Operators Python
123-chained-operators-python.md
tldr.md
124 Taxonomy On Sql
124-taxonomy-on-sql.md
tldr.md
125 The Weird Walrus
125-the-weird-walrus.md
tldr.md
126 Fully Persistent Arrays
126-fully-persistent-arrays.md
tldr.md
127 Persistent Data Structures Introduction
127-persistent-data-structures-introduction.md
tldr.md
128 Constant Folding Python
128-constant-folding-python.md
tldr.md
129 String Interning Python
129-string-interning-python.md
tldr.md
130 Recursion Visualizer Python
130-recursion-visualizer-python.md
tldr.md
131 Flajolet Martin
131-flajolet-martin.md
tldr.md
132 2q Cache
132-2q-cache.md
tldr.md
133 Israeli Queues
133-israeli-queues.md
tldr.md
134 1d Terrain
134-1d-terrain.md
tldr.md
135 Jaccard Minhash
135-jaccard-minhash.md
tldr.md
136 Ts Smoothing
136-ts-smoothing.md
tldr.md
137 Lfu
137-lfu.md
tldr.md
138 Morris Counter
138-morris-counter.md
tldr.md
139 Slowsort
139-slowsort.md
tldr.md
140 Bitcask
140-bitcask.md
tldr.md
141 Phi Accrual
141-phi-accrual.md
tldr.md
142 10x Engineer
142-10x-engineer.md
tldr.md
143 Decipher Repeated Key Xor
143-decipher-repeated-key-xor.md
tldr.md
144 Decipher Single Xor
144-decipher-single-xor.md
tldr.md
145 Python Iterable Integers
145-python-iterable-integers.md
tldr.md
146 Inheritance C
146-inheritance-c.md
tldr.md
147 Rum
147-rum.md
tldr.md
148 Consistent Hashing
148-consistent-hashing.md
tldr.md
149 Python Caches Integers
149-python-caches-integers.md
tldr.md
150 Fractional Cascading
150-fractional-cascading.md
tldr.md
151 Copy On Write
151-copy-on-write.md
tldr.md
152 Midpoint Insertion Caching Strategy
152-midpoint-insertion-caching-strategy.md
tldr.md
153 Fsm Python
153-fsm-python.md
tldr.md
154 Bayesian Average
154-bayesian-average.md
tldr.md
155 Sliding Window Ratelimiter
155-sliding-window-ratelimiter.md
tldr.md
156 Idf
156-idf.md
tldr.md
157 Better Programmer
157-better-programmer.md
tldr.md
158 Python Prompts
158-python-prompts.md
tldr.md
159 Rule 30 Cellular Automata
159-rule-30-cellular-automata.md
tldr.md
160 Function Overloading
160-function-overloading.md
tldr.md
161 Isolation Forest
161-isolation-forest.md
tldr.md
162 Image Steganography
162-image-steganography.md
tldr.md
163 Long Integers Python
163-long-integers-python.md
tldr.md
164 I Changed My Python
164-i-changed-my-python.md
tldr.md
165 Benchmark And Compare Pagination Approach In Mongodb
165-benchmark-and-compare-pagination-approach-in-mongodb.md
tldr.md
166 Mongodb Cursor Skip Is Slow
166-mongodb-cursor-skip-is-slow.md
tldr.md
167 Fast And Efficient Pagination In Mongodb
167-fast-and-efficient-pagination-in-mongodb.md
tldr.md
168 Making Http Requests Using Netcat
168-making-http-requests-using-netcat.md
tldr.md
Blum Blum Shub algorithm
Middle-square method
Lagged Fibonacci generator
Rule 30
Cellular Automaton
Mathematica

Cellular Automaton

Before we dive into Rule 30, we will spend some time understanding Cellular Automaton. A Cellular Automaton is a discrete model consisting of a regular grid, of any dimension, with each cell of the grid having a finite number of states and a neighborhood definition. There are rules that determine how these cells interact and transition into the next generation (state). The rules are mostly mathematical/programmable functions that depend on the current state of the cell and its neighborhood.

Cellular Automata

In the above Cellular Automaton, each cell has 2 finite states 0 (shown in red), 1 (shown in black). Each cell transitions into the next generation by XORing the state values of its 8 neighbors. The first generation (initial state) of the grid is allocated at random and the state transitions, of the entire grid, is as below

Cellular Automata Demo

Cellular Automata was originally conceptualized in the 1940s by Stanislaw Ulam and John von Neumann; it finds its application in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling. In the 1980s, Stephen Wolfram did a systematic study of one-dimensional cellular automata (also called elementary cellular automata) on which Rule 30 is based.

Rule 30

Rule 30 is an elementary (one-dimensional) cellular automaton where each cell has two possible states 0 (shown in red) and 1 (shown in black). The neighborhood of a cell is its two immediate neighbors, one on its left and other on right. The next state (generation) of the cell depends on its current state and the state of its neighbors; the transition rules are as illustrated below

Rule 30

The above transition rules could be simplified as left XOR (central OR right).

We visualize Rule 30 in a 2-dimensional grid where each row represents one generation (state). The next generation (state) of the cells is computed and populated in the row below. Each row contains a finite number of cells which “wraps around” at the end.

Rule 30 in action

The above pattern emerges from an initial state (row 0) in a single cell with state 1 (shown as black) surrounded by cells with state 0 (red). The next generation (as seen in row 1) is computed using the rule chart mentioned above. The vertical axis represents time and any horizontal cross-section of the image represents the state of all the cells in the array at a specific point in the pattern’s evolution.

Chaos in Rule 30

As the pattern evolves, frequent red triangles of varying sizes pop up but the structure as a whole has no recognizable pattern. The above snapshot of the grid was taken at a random point of time and we could observe chaos and aperiodicity. This property is exploited to generate pseudorandom numbers.

Pseudorandom Number Generation

As established earlier, Rule 30 is exhibits aperiodic and chaotic behavior and hence it produces complex, seemingly random patterns from simple, well-defined rules. To generate random numbers from using Rule 30 we use the center column and pick a batch of n random bits and form the required n bit random number from it. The next random number is built using the next n bits from the column.

Pseudorandom Number Rule 30

If we always start from the first row, the sequence of the numbers we generate will always be predictable - which is not what we want. To make things pseudorandom, we take a random seed value (ex: current timestamp) and skip that number of bits and then pick batches of n and build random numbers.

The pseudorandom numbers generated using Rule 30 are not cryptographically secure but are suitable for simulation as long as we do not use bad seed like 0.

One major advantage of using Rule 30 to generate pseudorandom numbers is that we could generate multiple random numbers in parallel by picking multiple columns to batch n bits each at random. A sample 8-bit random integer sequence generated using this method with seed 0 is 220, 197, 147, 174, 117, 97, 149, 171, 240, 241, etc.

The seed value could also be used as the initial state (row 0) for Rule 30 and random numbers are then simply the n bits batches picked from the center column starting from row 0. This approach is more efficient but is heavily dependent on the quality of seed value, as a bad seed value could make things extremely predictable. A demonstration of this approach could be found on Wolfram Cloud Demonstration Page.

Rule 30 in the real world

Rule 30 is also seen in nature, on the shell of code snail species Conus textile. The Cambridge North railway station is decorated with architectural panels displaying the evolution of Rule 30.

Conclusion

If you found Rule 30 interesting I urge you to write your own simulation of using p5 library; you could keep it generic enough to so that the program could generate patterns for different rules like 90, 110, 117, etc. The patterns generated using these rules are quite interesting. If you want, you could things to the next level and extend rule to work in 3 dimensions and see how patterns evolve. I believe programming is fun when it is visual.

It is exciting when two seemingly unrelated fields, Cellular Automata and Cryptography, come together and create something wonderful. Although this algorithm is not widely used anymore, because of more efficient algorithms, it urges us to be creative in using Cellular Automata in more ways than one. This article is first in the series of Cellular Automata, so stay tuned and watch this space for more.