2b:null 2c:["$","title",null,{"children":"The “DNA” of 4-valued FORMs | Peter Hofmann | Observable"}] 31:T564,md`

### Correspondence A FORM can have only one unique *formDNA* but a *formDNA* can map to a great variety of FORMs. This is most apparent for the simplest *formDNAs* that consist of only a single value, like ${showInl('::1')}, which can be ${showInl('()::1')}, ${showInl('((1))::1')}, ${showInl('{open|23}::1')} or any other FORM without variables that yields the marked state. In this case, the *formDNA* is identical to the mark. Both FORMs below share the same *formDNA*, because they are just two different expressions of the same FORM. Their [SelFis](https://www.carl-auer.de/magazin/systemzeit/how-does-system-function-operate) will be congruent: ${showInl('((a)b)::0303200223010000')} ${tex`\Leftrightarrow`} ${showInl('(((a)a)b(a))::0303200223010000')} But be careful, because different interpretation orders can yield different *formDNAs* (as we have already seen in the introduction): ${showInl('((a)b).[a,b]::0303200223010000')} ${tex`\nLeftrightarrow`} ${showInl('((a)b).[b,a]::0220303000003210')} Try plugging the FORM into my [SelFi-CA](https://observablehq.com/@formsandlines/1d-ca-for-4-valued-form-logic-selfis) and alternate between both variable orders, you'll see that the stripes will evolve in different directions. Of course, FORMs like \`\`(ab)\`\` are commutative and therefore invariant of interpretation ordering. `32:T519,md`

### Applications *formDNAs* can get pretty huge when we have FORMs with more than 3 variables: ${show('{a,b,c,d,e}')} You can still share them by copy & paste, plug them into a SelFi-CA and see what they look like. But you can also just take the FORM itself, right? My initial motivation behind this concept was practical, since I needed a way to work with value tables in my CAs (this took away the need to calculate the FORM directly over and over again) and quaternary numbers just translate nicely to binary, which is convenient for rule extraction by bitmasking (although there may still be more effective ways to do this, I am not an academic computer scientist). If you have played around with my [SelFi-CA](https://observablehq.com/@formsandlines/1d-ca-for-4-valued-form-logic-selfis) for a while, you may have noticed that you can directly manipulate the *formDNA* by either entering the whole string, randomly generating it or setting each “rule” individually. In this way, you can come up with FORMs that we most likely don't know any formal expression for. There is now also a function in *formform* that can generate random *formDNA* strings for a given variable number:`33:[] 2f:{"id":"80472c9935c36461","slug":"the-dna-of-4-valued-forms","title":"The “DNA” of 4-valued FORMs","thumbnail":"2ab5a96cdf75e50a3b456c5f9ee8521a760ec1c2d691f0a4f25979838ea11aea","access_level":"public","create_time":"$D2019-08-29T19:24:12.280Z","update_time":"$D2020-11-01T11:39:36.955Z","view_time":null,"first_public_time":"$D2019-09-29T14:09:40.294Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"e581d618919cd266","login":"formsandlines","name":"Peter Hofmann","avatar_url":"https://avatars-next.observableusercontent.com/da037e21011ef7f0553d73420763f4a2071c0bd9191552279dd8ebc154fc28b8","tier":"starter_2024"},"version":1078,"publish_version":1078,"role":null,"authored":false,"authors":[{"type":"individual","id":"b20948974f4dc365","login":"formsandlines","name":"Peter Hofmann","avatar_url":"https://avatars-next.observableusercontent.com/da037e21011ef7f0553d73420763f4a2071c0bd9191552279dd8ebc154fc28b8","tier":null}],"members":[{"type":"individual","id":"b20948974f4dc365","login":"formsandlines","name":"Peter Hofmann","avatar_url":"https://avatars-next.observableusercontent.com/da037e21011ef7f0553d73420763f4a2071c0bd9191552279dd8ebc154fc28b8","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"8fed57265de89c53","slug":"research","title":"Research","owner":{"id":"e581d618919cd266","login":"formsandlines","name":"Peter Hofmann"}},"type":"notebook","body":{"title":"The “DNA” of 4-valued FORMs","cells":[{"id":0,"value":"md`# The “DNA” of 4-valued FORMs`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":844,"value":"metadata({autor:\"Peter Hofmann\", created:\"29.08.2019\", fstPublish:\"29.09.2019\", thisPublish:\"01.11.2020\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":272,"value":"md`I propose a code format for [FORM logic](https://uformiform.info) that is an abstraction of the value table and is thus able to codify every possible FORM. At the same time, it is a quaternary number that can be used for practical applications like cellular automata.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":263,"value":"md`---\n\n### Introduction to *formDNA*\n\nA variable in a FORM can be interpreted by 4 possible values:
\n\\`\\`0\\`\\` or *unmarked*, \\`\\`1\\`\\` or *marked,* \\`\\`2\\`\\` or *undetermined,* \\`\\`3\\`\\` or *imaginary*\n\nWhen we call \\`\\`form.calcAll()\\`\\` in [formform](https://github.com/formsandlines/formform) to calculate all possible interpretations of a FORM, it will output a list of key-value pairs where the keys represent each interpretation for the variable set and the values are their corresponding results:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":338,"value":"formform.form.calcAll('((a)b)')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":416,"value":"md`Since we are using keys, the list entries don't have to be in a specific order, although the function I wrote for *formform* orders them in correspondance to the [quaternary numeral system](https://en.wikipedia.org/wiki/Quaternary_numeral_system) (e.g.: 00, 01, 02, 03, 10, 11, 12, 13, 20, …).\n\nThis “natural” algorithmic ordering can be used as a convention to abstract away from the interpretation details and string together the results. We will then obtain a *quaternary number* which I call *formDNA* because it is also a unique identifier for a specific equivalence class of (possibly infinite) FORMs that share the same value table:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":614,"value":"'::'+Object.values(formform.form.calcAll('((a)b)')).reverse().join('')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":872,"value":"md`As you can see, I reversed the order of the digits so that it can be parsed from right (with each variable at \\`\\`0/n\\`\\`) to left (with each variable at \\`\\`3/i\\`\\`) and I prefixed it with a double colon to distinguish *formDNAs* from other numbers.\n\nYou can also denote a FORM with its *formDNA* like this:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":868,"value":"show('((a)b)::0303200223010000')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":878,"value":"md`Or, if you need to specify the interpretation order of your variables:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":880,"value":"show('{L,E,R}{R,E,L}{L,R,E}',['L','E','R'])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":580,"value":"md`This would be unnecessary if we'd use a key-value approach as in \\`\\`form.calcAll()\\`\\`, but here we have to specify an interpretation order for our variables if we don't want it to be alphabetical (or to be precise: ordered by [UTF-16](https://en.wikipedia.org/wiki/UTF-16) code values, which is the default order that is implied in *formDNA*).\n\nFor example, ${showInl('{\"me\",\"you\"}::2303220323012202')} \nimplies that:
\\`\\`(\"me\":0,\"you\":0) = 2\\`\\`, \\`\\`(\"me\":0,\"you\":1) = 0\\`\\`, \\`\\`(\"me\":1,\"you\":0) = 1\\`\\`, and so on. [See for yourself](https://formform.formsandlines.eu/tricorder/#%7B%22me%22,%22you%22%7D#calc).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":604,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":436,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":979,"value":"random_dna = '::'+formform.dna.genRandomDNA(4)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":988,"value":"show(random_dna)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":999,"value":"md`If you are curious about the value distribution in a *formDNA*, I have created a visualization format for 4-valued FORMs called [*vmap*](https://observablehq.com/@formsandlines/recursive-mapping-of-4-valued-forms-with-vmaps) that is a direct recursive mapping of *formDNA* in 2 dimensions:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":983,"value":"html`${formform.dna.vmap(random_dna)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":975,"value":"md`It is obvious from the completeness of the FORM calculus that a formal expression for every possible *formDNA* can be derived (take a look at my notebook [“Determination in the undetermined”](https://observablehq.com/@formsandlines/determination-in-the-undetermined) to see how this can be done). But it may be cumbersome and the formula will be incredibly huge without simplification. This is why *formDNA* can be handy if you (for whatever reason) want to examine/compare exotic FORMs or “mutations” in a CA or if you want to generally denote a class of equivalent FORMs without having to refer to a specific FORMal expression.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1044,"value":"md`

\n\n### Implementation and usage\n\nYou can easily convert FORMs into *formDNA* by clicking the \\`\\`\\`formDNA\\`\\`\\` button in my [FORM tricorder](https://formform.formsandlines.eu/tricorder/). This can also be used with *formDNA* input to check if a string is valid *formDNA* (note that you always have to prepend \\`\\`\\`::\\`\\`\\` to mark a string as *formDNA*, otherwise it will just be parsed as a FORM).\n\nIf you want to do this in your own projects or Observable notebooks, *import* or *require* \\`\\`\\`formform\\`\\`\\` and use following function:\n\n~~~js\n formform.dna.formToDNA(input, varorder, options)\n~~~\n\nFurther details can be found on the [GitHub page](https://github.com/formsandlines/formform) of formform.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":706,"value":"md`---\n### Appendix`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":902,"value":"show = (input, varorder=undefined, options=undefined) => html`${formform.dna.formToDNA(input, varorder, {output: 'html'})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":946,"value":"showInl = (input, varorder=undefined) => html`${formform.dna.formToDNA(input, varorder, {output: 'text'})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":911,"value":"{\n return html`\n`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1026,"value":"typoCSS","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1021,"value":"LANG = 'en'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":147,"value":"formform = require('formform@0.9.2')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":840,"value":"import {metadata, typoCSS} with { LANG as setLang } from 'b537017f986f4d7c@307'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"