2d:null 2e:["$","title",null,{"children":"Tarski's World | Vinícius Mello | Observable"}] 33:Tb10,function tarski_world(n, d) { if (!Number.isInteger(n) || n <= 0) { throw new TypeError("n deve ser um inteiro positivo."); } const cell = 70; const padding = 20; const boardSize = n * cell; const width = boardSize + padding * 2; const height = boardSize + padding * 2; const objects = normalizeObjects(d); const gridLines = []; for (let i = 0; i <= n; i += 1) { const v = padding + i * cell; gridLines.push( `` ); gridLines.push( `` ); } const coordsLabels = []; for (let i = 1; i <= n; i += 1) { const x = padding + (i - 0.5) * cell; const y = padding + (n - i + 0.5) * cell; coordsLabels.push( `${i}` ); coordsLabels.push( `${i}` ); } const objectShapes = objects.map((obj) => { validateObject(obj, n); const [col, row] = obj.pos; const cx = padding + (col - 0.5) * cell; const cy = padding + (n - row + 0.5) * cell; const fill = obj.color || "#1f77b4"; const stroke = "#111"; const label = escapeXml(obj.name || ""); const type = String(obj.type || "circle").toLowerCase(); let shapeSvg = ""; if (type === "triangle") { const h = 18; const pts = [ `${cx},${cy - h}`, `${cx - h},${cy + h}`, `${cx + h},${cy + h}` ].join(" "); shapeSvg = ``; } else if (type === "square") { const s = 34; shapeSvg = ``; } else { const r = 18; shapeSvg = ``; } const nameSvg = label ? `${label}` : ""; return `${shapeSvg}${nameSvg}`; }); let r = [ `` ].join("\n"); return html`${r}`; }34:T137b,function fol_parser(str) { if (typeof str !== "string") { throw new TypeError("fol_parser espera uma string."); } const input = str; const tokens = tokenize(input); let pos = 0; const boundScopes = [new Set()]; function current() { return tokens[pos]; } function at(type, value) { const tok = current(); if (!tok) return false; if (type && tok.type !== type) return false; if (value !== undefined && tok.value !== value) return false; return true; } function consume(type, value) { const tok = current(); if ( !tok || (type && tok.type !== type) || (value !== undefined && tok.value !== value) ) { const expected = value !== undefined ? `${type}(${value})` : type; const got = tok ? `${tok.type}(${tok.value})` : "EOF"; throw syntaxError( `Esperado ${expected}, mas encontrado ${got}.`, tok ? tok.index : input.length ); } pos += 1; return tok; } function syntaxError(message, index) { const pointer = `${input}\n${" ".repeat(Math.max(0, index))}^`; return new SyntaxError(`${message}\n${pointer}`); } function pushScope() { boundScopes.push(new Set(boundScopes[boundScopes.length - 1])); } function popScope() { boundScopes.pop(); } function bindVariable(name) { boundScopes[boundScopes.length - 1].add(name); } function isBound(name) { return boundScopes[boundScopes.length - 1].has(name); } function parseFormula() { return parseIff(); } function parseIff() { let left = parseImplies(); while (at("SYMBOL", "↔")) { const op = consume("SYMBOL", "↔").value; const right = parseImplies(); left = { type: "Binary", operator: op, left, right }; } return left; } function parseImplies() { let left = parseOr(); while (at("SYMBOL", "→")) { const op = consume("SYMBOL", "→").value; const right = parseOr(); left = { type: "Binary", operator: op, left, right }; } return left; } function parseOr() { let left = parseAnd(); while (at("SYMBOL", "∨")) { const op = consume("SYMBOL", "∨").value; const right = parseAnd(); left = { type: "Binary", operator: op, left, right }; } return left; } function parseAnd() { let left = parseUnary(); while (at("SYMBOL", "∧")) { const op = consume("SYMBOL", "∧").value; const right = parseUnary(); left = { type: "Binary", operator: op, left, right }; } return left; } function parseUnary() { if (at("SYMBOL", "¬")) { consume("SYMBOL", "¬"); return { type: "Not", operand: parseUnary() }; } if (at("SYMBOL", "∀") || at("SYMBOL", "∃")) { const quantifier = consume("SYMBOL").value; const variable = consume("IDENT").value; if (at("SYMBOL", ".") || at("SYMBOL", ":")) { consume("SYMBOL"); } pushScope(); bindVariable(variable); const body = parseUnary(); popScope(); return { type: "Quantifier", quantifier, variable, body }; } return parsePrimary(); } function parsePrimary() { if (at("SYMBOL", "(")) { consume("SYMBOL", "("); const expr = parseFormula(); consume("SYMBOL", ")"); return expr; } return parseAtomic(); } function parseAtomic() { const leftTerm = parseTerm(); if (at("SYMBOL", "=")) { consume("SYMBOL", "="); const rightTerm = parseTerm(); return { type: "Equality", left: leftTerm, right: rightTerm }; } if (at("SYMBOL", "≠")) { consume("SYMBOL", "≠"); const rightTerm = parseTerm(); return { type: "Not", operand: { type: "Equality", left: leftTerm, right: rightTerm } }; } if (leftTerm.type === "Function") { return { type: "Predicate", name: leftTerm.name, args: leftTerm.args }; } if (leftTerm.type === "Constant") { return { type: "Predicate", name: leftTerm.name, args: [] }; } throw syntaxError( "Fórmula atômica inválida. Esperado predicado ou igualdade.", current() ? current().index : input.length ); } function parseTerm() { const ident = consume("IDENT").value; if (at("SYMBOL", "(")) { consume("SYMBOL", "("); const args = []; if (!at("SYMBOL", ")")) { args.push(parseTerm()); while (at("SYMBOL", ",")) { consume("SYMBOL", ","); args.push(parseTerm()); } } consume("SYMBOL", ")"); return { type: "Function", name: ident, args }; } if (isBound(ident) || /^[xyzuvw][A-Za-z0-9_]*$/.test(ident)) { return { type: "Variable", name: ident }; } return { type: "Constant", name: ident }; } const ast = parseFormula(); if (!at("EOF")) { const tok = current(); throw syntaxError(`Token inesperado: ${tok.value}`, tok.index); } return ast; }35:T934,function evalPredicate(name, args, world) { const pred = String(name); if (pred === "Triangle") { expectArity(pred, args, 1); return getObj(args[0], world).type === "triangle"; } if (pred === "Square") { expectArity(pred, args, 1); return getObj(args[0], world).type === "square"; } if (pred === "Circle") { expectArity(pred, args, 1); return getObj(args[0], world).type === "circle"; } if (pred === "Red") { expectArity(pred, args, 1); return getObj(args[0], world).color === "red"; } if (pred === "Blue") { expectArity(pred, args, 1); return getObj(args[0], world).color === "blue"; } if (pred === "Gray") { expectArity(pred, args, 1); return getObj(args[0], world).color === "gray"; } if (pred === "AboveOf") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.y > y.y; } if (pred === "BelowOf") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.y < y.y; } if (pred === "SameRow") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.y === y.y; } if (pred === "SameColumn") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.x === y.x; } if (pred === "SameColor") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.color === y.color; } if (pred === "SameType") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.type === y.type; } if (pred === "LeftOf") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.x < y.x; } if (pred === "RightOf") { expectArity(pred, args, 2); const x = getObj(args[0], world); const y = getObj(args[1], world); return x.x > y.x; } if (pred === "Between") { expectArity(pred, args, 3); const x = getObj(args[0], world); const y = getObj(args[1], world); const z = getObj(args[2], world); return isBetween(x, y, z); } throw new Error(`Predicado não suportado: ${pred}`); }36:T865,function evalFormula(node, world, env) { if (!node || typeof node !== "object") { throw new TypeError("AST inválida."); } switch (node.type) { case "Not": return !evalFormula(node.operand, world, env); case "Binary": { if (node.operator === "∧") return ( evalFormula(node.left, world, env) && evalFormula(node.right, world, env) ); if (node.operator === "∨") return ( evalFormula(node.left, world, env) || evalFormula(node.right, world, env) ); if (node.operator === "→") return ( !evalFormula(node.left, world, env) || evalFormula(node.right, world, env) ); if (node.operator === "↔") { const left = evalFormula(node.left, world, env); const right = evalFormula(node.right, world, env); return left === right; } throw new Error(`Operador lógico não suportado: ${node.operator}`); } case "Quantifier": { const variable = node.variable; if (node.quantifier === "∀") { for (const name of world.names) { const nextEnv = Object.create(env); nextEnv[variable] = name; if (!evalFormula(node.body, world, nextEnv)) { return false; } } return true; } if (node.quantifier === "∃") { for (const name of world.names) { const nextEnv = Object.create(env); nextEnv[variable] = name; if (evalFormula(node.body, world, nextEnv)) { return true; } } return false; } throw new Error(`Quantificador não suportado: ${node.quantifier}`); } case "Equality": { const left = evalTerm(node.left, world, env); const right = evalTerm(node.right, world, env); return left === right; } case "Predicate": { const args = (node.args || []).map((arg) => evalTerm(arg, world, env)); return evalPredicate(node.name, args, world); } default: throw new Error(`Nó de fórmula não suportado: ${node.type}`); } }37:T440,function buildWorld(d) { if (!d || typeof d !== "object") { throw new TypeError("d deve ser um objeto com os elementos do mundo."); } const entries = Array.isArray(d) ? d.map((obj, i) => [obj && obj.name ? String(obj.name) : `o${i + 1}`, obj]) : Object.entries(d); const objects = new Map(); for (const [name, raw] of entries) { if (!raw || typeof raw !== "object") { throw new TypeError(`Objeto inválido para ${name}.`); } const type = String(raw.type || "").toLowerCase(); const color = String(raw.color || "").toLowerCase(); const pos = raw.pos; if (!Array.isArray(pos) || pos.length !== 2) { throw new TypeError(`Objeto ${name} deve ter pos: [x, y].`); } const [x, y] = pos; if (!Number.isInteger(x) || !Number.isInteger(y)) { throw new TypeError( `Objeto ${name} deve ter coordenadas inteiras em pos.` ); } objects.set(String(name), { name: String(name), type, color, x, y }); } return { names: [...objects.keys()], objects }; }38:[] 31:{"id":"22cc6a987c574f36","slug":"tarskis-world","title":"Tarski's World","thumbnail":"c7492a17e8496aa94bedc2f4fbebbe045230ee9f5743815394b6ac4afdac35b0","access_level":"public","create_time":"$D2026-04-16T13:16:46.338Z","update_time":"$D2026-04-17T01:48:26.548Z","view_time":null,"first_public_time":"$D2026-04-16T15:25:10.934Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"7a4ecd430a93cd6f","login":"vinicius-mello","name":"Vinícius Mello","avatar_url":"https://avatars-next.observableusercontent.com/b5d4671aa488d8bcb984f575f985332403cde5f63942ea883a6a1a630941ec05","tier":"starter_2024"},"version":125,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"ecc4854ab7ad95b6","login":"vinicius-mello","name":"Vinícius Mello","avatar_url":"https://avatars-next.observableusercontent.com/b5d4671aa488d8bcb984f575f985332403cde5f63942ea883a6a1a630941ec05","tier":null}],"members":[{"type":"team","id":"7a4ecd430a93cd6f","login":"vinicius-mello","name":"Vinícius Mello","avatar_url":"https://avatars-next.observableusercontent.com/b5d4671aa488d8bcb984f575f985332403cde5f63942ea883a6a1a630941ec05","tier":"starter_2024","role":"editor"},{"type":"individual","id":"ecc4854ab7ad95b6","login":"vinicius-mello","name":"Vinícius Mello","avatar_url":"https://avatars-next.observableusercontent.com/b5d4671aa488d8bcb984f575f985332403cde5f63942ea883a6a1a630941ec05","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Tarski's World","cells":[{"id":0,"value":"# Tarski's World","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"tarski_world(5, world)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"viewof formula = Inputs.text({\n label: \"Formula\",\n width: 600,\n value: \"∀x(Circle(x)→AboveOf(x,f))\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":65,"value":"eval_fol_tarski(fol_parser(formula), world)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"Symbols: ¬, ∧, ∨, ∀, ∃, →, ↔, =, ≠\n\nPredicates: Red(x), Blue(x), Gray(x), Triangle(x), Square(x), Circle(x),\nAboveOf(x,y), BelowOf(x,y), RightOf(x,y), LeftOf(x,y), Between(x,y,z),\nSameColor(x,y),SameType(x,y),SameRow(x,y),SameColumn(x,y)\n\nExamples\n - ∀x(Circle(x)→AboveOf(x,f))\n - ∃x(Square(x)∧Red(x))\n - ∀x(Circle(x)→(∃y(Square(y)∧SameColor(x,y))))\n - ∃x(Square(x)∧(∀y(Triangle(y)→RightOf(x,y))))\n - ∀x(∃y(SameColumn(x,y)∧x≠y))","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":61,"value":"world = ({\n a: { type: \"circle\", color: \"blue\", pos: [3, 5] },\n b: { type: \"circle\", color: \"gray\", pos: [1, 4] },\n c: { type: \"circle\", color: \"blue\", pos: [2, 4] },\n d: { type: \"triangle\", color: \"red\", pos: [4, 4] },\n e: { type: \"square\", color: \"red\", pos: [1, 3] },\n f: { type: \"triangle\", color: \"gray\", pos: [3, 3] },\n g: { type: \"square\", color: \"gray\", pos: [5, 3] },\n h: { type: \"square\", color: \"gray\", pos: [2, 2] },\n i: { type: \"triangle\", color: \"gray\", pos: [4, 1] },\n j: { type: \"square\", color: \"blue\", pos: [5, 1] }\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"function escapeXml(text) {\n return String(text)\n .replace(/&/g, \"&\")\n .replace(//g, \">\")\n .replace(/\\\"/g, \""\")\n .replace(/'/g, \"'\");\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"function normalizeObjects(d) {\n if (!d || typeof d !== \"object\") {\n throw new TypeError(\"d deve ser um objeto ou array de objetos.\");\n }\n\n if (Array.isArray(d)) {\n return d.map((obj, i) => ({\n name: obj && obj.name ? String(obj.name) : `o${i + 1}`,\n type: obj && obj.type,\n color: obj && obj.color,\n pos: obj && obj.pos\n }));\n }\n\n return Object.entries(d).map(([name, obj]) => ({\n name,\n type: obj && obj.type,\n color: obj && obj.color,\n pos: obj && obj.pos\n }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"function validateObject(obj, n) {\n if (!obj || !Array.isArray(obj.pos) || obj.pos.length !== 2) {\n throw new TypeError(\n `Objeto ${\n obj && obj.name ? obj.name : \"\"\n } deve ter pos: [x, y].`\n );\n }\n\n const [x, y] = obj.pos;\n if (!Number.isInteger(x) || !Number.isInteger(y)) {\n throw new TypeError(`Coordenadas de ${obj.name} devem ser inteiras.`);\n }\n\n if (x < 1 || x > n || y < 1 || y > n) {\n throw new RangeError(\n `Coordenadas de ${obj.name} fora do tabuleiro ${n}x${n}: [${x}, ${y}].`\n );\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"function tokenize(input) {\n const tokens = [];\n let i = 0;\n\n const singleCharSymbols = new Set([\n \"(\",\n \")\",\n \",\",\n \"=\",\n \".\",\n \":\",\n \"¬\",\n \"∧\",\n \"∨\",\n \"∀\",\n \"∃\",\n \"→\",\n \"↔\",\n \"≠\"\n ]);\n\n while (i < input.length) {\n const ch = input[i];\n\n if (/\\s/.test(ch)) {\n i += 1;\n continue;\n }\n\n if (singleCharSymbols.has(ch)) {\n tokens.push({ type: \"SYMBOL\", value: ch, index: i });\n i += 1;\n continue;\n }\n\n if (/[A-Za-z_]/.test(ch)) {\n const start = i;\n i += 1;\n while (i < input.length && /[A-Za-z0-9_]/.test(input[i])) {\n i += 1;\n }\n const ident = input.slice(start, i);\n tokens.push({ type: \"IDENT\", value: ident, index: start });\n continue;\n }\n\n throw new SyntaxError(`Caractere inválido '${ch}' na posição ${i}.`);\n }\n\n tokens.push({ type: \"EOF\", value: \"\", index: input.length });\n\n return tokens;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"function isBetween(a, b, c) {\n if ((a.x === b.x && a.y === b.y) || (c.x === b.x && c.y === b.y)) {\n return false;\n }\n\n const abx = b.x - a.x;\n const aby = b.y - a.y;\n const acx = c.x - a.x;\n const acy = c.y - a.y;\n\n const cross = abx * acy - aby * acx;\n if (cross !== 0) {\n return false;\n }\n\n const minX = Math.min(a.x, c.x);\n const maxX = Math.max(a.x, c.x);\n const minY = Math.min(a.y, c.y);\n const maxY = Math.max(a.y, c.y);\n\n return (\n (b.x > minX && b.x < maxX && b.y > minY && b.y < maxY) ||\n (a.x === c.x && b.x === a.x && b.y > minY && b.y < maxY) ||\n (a.y === c.y && b.y === a.y && b.x > minX && b.x < maxX)\n );\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"function getObj(name, world) {\n const obj = world.objects.get(name);\n if (!obj) {\n throw new Error(`Objeto '${name}' não encontrado no mundo.`);\n }\n return obj;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":37,"value":"function expectArity(name, args, arity) {\n if (args.length !== arity) {\n throw new Error(\n `Predicado ${name} espera ${arity} argumentos, recebeu ${args.length}.`\n );\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"function evalTerm(term, world, env) {\n switch (term.type) {\n case \"Variable\": {\n const bound = env[term.name];\n if (!bound) {\n throw new Error(`Variável livre sem valor: ${term.name}`);\n }\n return bound;\n }\n\n case \"Constant\": {\n if (!world.objects.has(term.name)) {\n throw new Error(`Constante '${term.name}' não está no mundo d.`);\n }\n return term.name;\n }\n\n case \"Function\":\n throw new Error(\n `Funções não são suportadas em eval_fol_tarski: ${term.name}`\n );\n\n default:\n throw new Error(`Termo não suportado: ${term.type}`);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":48,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"function eval_fol_tarski(ast, d) {\n const world = buildWorld(d);\n return evalFormula(ast, world, Object.create(null));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W38","stdlib":"1","titling":"auto"}} 32:"$31:body"