abstract-syntax-exercise.ipynb (56873B)
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1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 | { "cells": [ { "cell_type": "markdown", "id": "25085ca3", "metadata": {}, "source": [ "Before you turn this problem in, make sure everything runs as expected. First, **restart the kernel** (in the menubar, select Kernel $\\rightarrow$ Restart) and then **run all cells** (in the menubar, select Cell $\\rightarrow$ Run All).\n", "\n", "Make sure you fill in any place that says `???`, `YOUR CODE HERE`, \"???\", \"YOUR ANSWER HERE\", as well as your name and collaborators below:" ] }, { "cell_type": "code", "execution_count": 19, "id": "746e58f4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[36mNAME\u001b[39m: \u001b[32mString\u001b[39m = \u001b[32m\"Jack Sangdahl\"\u001b[39m\n", "\u001b[36mCOLLABORATORS\u001b[39m: \u001b[32mString\u001b[39m = \u001b[32m\"N/A\"\u001b[39m" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "val NAME = \"Jack Sangdahl\"\n", "val COLLABORATORS = \"N/A\"" ] }, { "cell_type": "markdown", "id": "bdb1d285", "metadata": {}, "source": [ "---" ] }, { "cell_type": "markdown", "id": "7428aa62-a2f8-45d6-a04d-568bbec79950", "metadata": {}, "source": [ "# Exercise: Syntax\n", "\n", "<!-- 3 Expressions -->\n", "\n", "<!-- 4 Binding and Scope -->\n", "\n", "<!-- 8 Recursion -->\n", "\n", "<!-- 9 Inductive Data Types -->\n", "\n", "<!-- 11 Concrete Syntax -->\n", "\n", "<!-- 12 Abstract Syntax and Parsing -->\n", "\n", "<!-- 13 Exercise: Syntax -->\n", "\n", "<!-- 14 Static Scoping -->\n", "\n", "### Learning Goals\n", "\n", "The primary learning goals of this assignment are to build intuition for\n", "the following:\n", "\n", "- working with abstract syntax trees;\n", "- how grammars are used to specify the syntax of programming\n", " languages; and\n", "- the distinction between concrete and abstract syntax.\n", "\n", "### Instructions\n", "\n", "This assignment asks you to write Scala code. There are restrictions\n", "associated with how you can solve these problems. Please pay careful\n", "heed to those. If you are unsure, ask the course staff.\n", "\n", "Note that `???` indicates that there is a missing function or code\n", "fragment that needs to be filled in. Make sure that you remove the `???`\n", "and replace it with the answer.\n", "\n", "Use the test cases provided to test your implementations. You are also\n", "encouraged to write your own test cases to help debug your work.\n", "However, please delete any extra cells you may have created lest they\n", "break an autograder.\n", "\n", "### Imports" ] }, { "cell_type": "code", "execution_count": 20, "id": "0c85487c", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "afc1ca0eb6ddf4e8f942a9b59b4d2e8e", "grade": false, "grade_id": "testing", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[32mimport \u001b[39m\u001b[36mRun this cell FIRST before testing.\n", "import $ivy.$ , org.scalatest._, events._, flatspec._\n", "\u001b[39m\n", "defined \u001b[32mfunction\u001b[39m \u001b[36mreport\u001b[39m\n", "defined \u001b[32mfunction\u001b[39m \u001b[36massertPassed\u001b[39m\n", "defined \u001b[32mfunction\u001b[39m \u001b[36mpassed\u001b[39m\n", "defined \u001b[32mfunction\u001b[39m \u001b[36mtest\u001b[39m" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "// Run this cell FIRST before testing.\n", "import $ivy.`org.scalatest::scalatest:3.2.19`, org.scalatest._, events._, flatspec._\n", "def report(suite: Suite): Unit = suite.execute(stats = true)\n", "def assertPassed(suite: Suite): Unit =\n", " suite.run(None, Args(new Reporter {\n", " def apply(e: Event) = e match {\n", " case e @ (_: TestFailed) => assert(false, s\"${e.message} (${e.testName})\")\n", " case _ => ()\n", " }\n", " }))\n", "def passed(points: Int): Unit = {\n", " require(points >=0)\n", " if (points == 1) println(\"*** 🎉 Tests Passed (1 point) ***\")\n", " else println(s\"*** 🎉 Tests Passed ($points points) ***\")\n", "}\n", "def test(suite: Suite, points: Int): Unit = {\n", " report(suite)\n", " assertPassed(suite)\n", " passed(points)\n", "}" ] }, { "cell_type": "code", "execution_count": 21, "id": "213c5736", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "3a42711f30a92ef63bef70549eb85bd2", "grade": false, "grade_id": "parser", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[32mimport \u001b[39m\u001b[36m$ivy.$\u001b[39m" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "// Run this cell FIRST before building your parser.\n", "import $ivy.`org.scala-lang.modules::scala-parser-combinators:2.4.0`" ] }, { "cell_type": "markdown", "id": "7e8272a2-d82f-4888-92d0-25fd120fd768", "metadata": {}, "source": [ "## Abstract Syntax Trees\n", "\n", "Let us consider the (abstract) syntax of a language of Boolean\n", "expressions:\n", "\n", "$$\\begin{array}{lrrl}\n", " \\text{Boolean expressions} & e& \\mathrel{::=}& x\n", " \\mid\\neg e_1\n", " \\mid e_1 \\wedge e_2\n", " \\mid e_1 \\vee e_2 \\\\\n", " \\text{variables} & x\n", "\\end{array}$$\n", "\n", "### Defining an Inductive Data Type\n", "\n", "We can write explicitly the above grammar with abstract syntax tree\n", "nodes:\n", "\n", "$$\\begin{array}{rrrl}\n", "\\text{Boolean expressions $\\texttt{BExpr}$} & e& \\mathrel{::=}&\n", " \\texttt{Var(} x\\texttt{)} \\\\\n", " & & \\mid& \\texttt{Not(}e_1\\texttt{)} \\\\\n", " & & \\mid& \\texttt{And(}e_1\\texttt{,}\\;e_2\\texttt{)} \\\\\n", " & & \\mid& \\texttt{Or(}e_1\\texttt{,}\\;e_2\\texttt{)} \\\\\n", "\\end{array}$$\n", "\n", "<span class=\"theorem-title\">**Exercise 1 (5 points)**</span> Define an\n", "inductive data type `BExpr` in Scala following the above grammar. Use\n", "the the names given above for the constructors, and use the Scala type\n", "`String` to represent variable names.\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 22, "id": "c17f7690", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "4fa8ce605b81e3837cc6f0a5eb8c87c0", "grade": false, "grade_id": "bexpr_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mtrait\u001b[39m \u001b[36mBExpr\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mVar\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mNot\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mAnd\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mOr\u001b[39m" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sealed trait BExpr\n", "case class Var(x: String) extends BExpr\n", "case class Not(e: BExpr) extends BExpr\n", "case class And(e1: BExpr, e2: BExpr) extends BExpr\n", "case class Or(e1: BExpr, e2: BExpr) extends BExpr" ] }, { "cell_type": "markdown", "id": "6bb509ab-c5d2-4069-b322-a4455dc8528b", "metadata": {}, "source": [ "#### Tests" ] }, { "cell_type": "code", "execution_count": 23, "id": "e0b0d389", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "9d5a59b48e6c646390a761b3e2129439", "grade": true, "grade_id": "bexpr_tests", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "*** 🎉 Tests Passed (5 points) ***\n" ] }, { "data": { "text/plain": [ "\u001b[36ma\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m)\n", "\u001b[36mb\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m)\n", "\u001b[36mc\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mVar\u001b[39m(x = \u001b[32m\"C\"\u001b[39m)\n", "\u001b[36mnot_b\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m))\n", "\u001b[36mnot_a\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m))\n", "\u001b[36me1\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mAnd\u001b[39m(e1 = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m), e2 = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m)))\n", "\u001b[36me2\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mAnd\u001b[39m(e1 = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m), e2 = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m)))\n", "\u001b[36me3\u001b[39m: \u001b[32mBExpr\u001b[39m = \u001b[33mOr\u001b[39m(\n", " e1 = \u001b[33mAnd\u001b[39m(e1 = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m), e2 = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m))),\n", " e2 = \u001b[33mAnd\u001b[39m(e1 = \u001b[33mVar\u001b[39m(x = \u001b[32m\"B\"\u001b[39m), e2 = \u001b[33mNot\u001b[39m(e = \u001b[33mVar\u001b[39m(x = \u001b[32m\"A\"\u001b[39m)))\n", ")" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "val a: BExpr = Var(\"A\")\n", "val b: BExpr = Var(\"B\")\n", "val c: BExpr = Var(\"C\")\n", "val not_b: BExpr = Not(b)\n", "val not_a: BExpr = Not(a)\n", "val e1: BExpr = And(a,not_b)\n", "val e2: BExpr = And(b, not_a)\n", "val e3: BExpr = Or(e1,e2)\n", "passed(5)" ] }, { "cell_type": "markdown", "id": "f1a9c60a-f0d8-4ce7-9de1-e5a9bb0a5cc0", "metadata": {}, "source": [ "### Converting to Negation Normal Form\n", "\n", "<span class=\"theorem-title\">**Exercise 2 (10 points)**</span> Write a\n", "function `nnf` to convert Boolean formulas from the above grammar to\n", "their Negation Normal Form (NNF). A Boolean formula is said to be in NNF\n", "iff the negation operator $\\neg$ (i.e., $\\texttt{Not}$) is only applied\n", "to the variables. For example, the following formula is in NNF because\n", "negation is only applied to $A$ or $B$, or both of which are variables:\n", "\n", "$$\n", "(A \\wedge \\neg B) \\vee (B \\land \\neg A) \\;.\n", "$$\n", "\n", "On the other hand, the following formula is not in NNF as negation is\n", "applied to a complex expression:\n", "\n", "$$\n", "\\neg(A \\vee \\neg B) \\wedge (\\neg B \\vee C) \\;.\n", "$$\n", "\n", "The negation normal form of the above formula is as follows:\n", "\n", "$$\n", "(\\neg A \\wedge \\neg B) \\wedge (\\neg B \\vee C) \\;.\n", "$$\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 24, "id": "1d442d89", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "72a0f6bf19b244521a77d1989ad162d4", "grade": false, "grade_id": "nnf_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mfunction\u001b[39m \u001b[36mnnf\u001b[39m" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def nnf(e: BExpr): BExpr = e match {\n", " case Var(_) => e // variable already in nnf\n", " case Not(Var(x)) => Not(Var(x)) // negated variable in nnf\n", " case Not(Not(e1)) => nnf(e1) // double negation cancels out\n", " case Not(And(e1, e2)) => Or(nnf(Not(e1)), nnf(Not(e2))) // apply de morgan law for and\n", " case Not(Or(e1, e2)) => And(nnf(Not(e1)), nnf(Not(e2))) // apply de morgan law for or\n", " case And(e1, e2) => And(nnf(e1), nnf(e2)) // recursively convert sub expression nnf\n", " case Or(e1, e2) => Or(nnf(e1), nnf(e2)) // recursively convert sub expression nnf\n", " case Not(e1) => Not(nnf(e1)) // handle negation in other cases\n", "}" ] }, { "cell_type": "markdown", "id": "adb8d1a9-1114-46c5-ac33-7b5269666896", "metadata": {}, "source": [ "#### Notes\n", "\n", "In general, a formula can be converted to NNF by applying three rules:\n", "\n", "Rule 1 \n", "Double negation is cancelled: $\\neg(\\neg e) = e$ for any Boolean formula\n", "$e$.\n", "\n", "Rule 2 \n", "De Morgan’s Law for conjunction:\n", "$\\neg(e_1 \\wedge e_2) = \\neg e_1 \\vee \\neg e_2$ for any two Boolean\n", "formulas $e_1$ and $e_2$.\n", "\n", "Rule 3 \n", "De Morgan’s Law for disjunction:\n", "$\\neg(e_1 \\vee e_2) = \\neg e_1 \\wedge \\neg e_2$ for any two Boolean\n", "formulas $e_1$ and $e_2$.\n", "\n", "The `nnf` function will have a case for each of the above rule. In\n", "addition, the function will also need to handle the following cases:\n", "\n", "- A variable $x$ or its negation $\\neg x$ is already in NNF.\n", "- An expression $e_1 \\wedge e_2$ is in NNF iff $e_1$ and $e_2$ are in\n", " NNF.\n", "- An expression $e_1 \\vee e_2$ is in NNF iff $e_1$ and $e_2$ are in\n", " NNF.\n", "- An expression $\\neg e_1$ is in NNF iff $e_1$ is in NNF and none of\n", " the above 3 rules apply.\n", "\n", "If you handle all the 7 cases described above, then your `nnf` function\n", "will likely be correct, though of course, you will want to test it.\n", "\n", "#### Tests" ] }, { "cell_type": "code", "execution_count": 25, "id": "70af5768", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "388fcf4fd08defa19603710f164045cb", "grade": true, "grade_id": "nnf_tests", "locked": true, "points": 10, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[36mRun starting. Expected test count is: 7\u001b[0m\n", "\u001b[32mcmd25$Helper$$anon$1:\u001b[0m\n", "\u001b[32mnnf\u001b[0m\n", "\u001b[32m- should Test Case 1\u001b[0m\n", "\u001b[32m- should Test Case 2\u001b[0m\n", "\u001b[32m- should Test Case 3\u001b[0m\n", "\u001b[32m- should Test Case 4\u001b[0m\n", "\u001b[32m- should Test Case 5\u001b[0m\n", "\u001b[32m- should Test Case 6\u001b[0m\n", "\u001b[32m- should Test Case 7\u001b[0m\n", "\u001b[36mRun completed in 2 milliseconds.\u001b[0m\n", "\u001b[36mTotal number of tests run: 7\u001b[0m\n", "\u001b[36mSuites: completed 1, aborted 0\u001b[0m\n", "\u001b[36mTests: succeeded 7, failed 0, canceled 0, ignored 0, pending 0\u001b[0m\n", "\u001b[32mAll tests passed.\u001b[0m\n", "*** 🎉 Tests Passed (10 points) ***\n" ] } ], "source": [ "test(new AnyFlatSpec {\n", " val testcase1 = Var(\"A\")\n", " val testcase2 = Not(Not(Var(\"B\")))\n", " val testcase3 = And(Var(\"A\"), Or(Var(\"B\"), Var(\"C\")))\n", " val testcase4 = Or(Var(\"A\"), And(Var(\"B\"), Var(\"C\")))\n", " val testcase5 = Not(And(Var(\"A\"), Var(\"B\")))\n", " val testcase6 = Not(Or(Var(\"A\"), Var(\"B\")))\n", " val testcase7 = Or(And(testcase5,testcase6),testcase2)\n", "\n", " \"nnf\" should \"Test Case 1\" in { assertResult( Var(\"A\") ) { nnf(testcase1)} }\n", " it should \"Test Case 2\" in { assertResult( Var(\"B\") ) { nnf(testcase2) } }\n", " it should \"Test Case 3\" in { assertResult( And(Var(\"A\"), Or(Var(\"B\"), Var(\"C\"))) ) { nnf(testcase3) } }\n", " it should \"Test Case 4\" in { assertResult( Or(Var(\"A\"), And(Var(\"B\"), Var(\"C\"))) ) { nnf(testcase4) } }\n", " it should \"Test Case 5\" in { assertResult( Or(Not(Var(\"A\")), Not(Var(\"B\"))) ) { nnf(testcase5) } }\n", " it should \"Test Case 6\" in { assertResult( And(Not(Var(\"A\")), Not(Var(\"B\"))) ) { nnf(testcase6) } }\n", " it should \"Test Case 7\" in { assertResult( Or(And(Or(Not(Var(\"A\")), Not(Var(\"B\"))), And(Not(Var(\"A\")), Not(Var(\"B\")))), Var(\"B\")) ) { nnf(testcase7) } }\n", "}, 10)" ] }, { "cell_type": "markdown", "id": "2e45d4e9-4d1a-474b-a3ae-115bd54edb56", "metadata": {}, "source": [ "### Substitution\n", "\n", "<span class=\"theorem-title\">**Exercise 3 (5 points)**</span> Write a\n", "function `subst` that substitutes *in* a given Boolean expression *with*\n", "another expression *for* a given variable name. For example,\n", "substituting in $$\n", "\\neg(A \\vee B) \\wedge (\\neg B \\vee C)\n", "$$ with $D$ for $B$ yields $$\n", "\\neg(A \\vee D) \\wedge (\\neg D \\vee C) \\;.\n", "$$\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 26, "id": "e4c5dda3", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "1f82f78050d59cbb7dd28d88085f4b63", "grade": false, "grade_id": "subst_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mfunction\u001b[39m \u001b[36msubst\u001b[39m" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def subst(in_e: BExpr, with_e: BExpr, for_x: String): BExpr = in_e match {\n", " case Var(x) if x == for_x => with_e // variable matches? substitute with with_e\n", " case Var(x) => Var(x) // if variable does not match, leavh unchanged\n", " case Not(e1) => Not(subst(e1, with_e, for_x)) // recursively apply substitution inside not\n", " case And(e1, e2) => And(subst(e1, with_e, for_x), subst(e2, with_e, for_x)) // recursively apply substitution inside and\n", " case Or(e1, e2) => Or(subst(e1, with_e, for_x), subst(e2, with_e, for_x)) // recursively apply substitution inside Or\n", "}" ] }, { "cell_type": "markdown", "id": "f454f19f-fb7c-4eeb-807f-7c3e39419d67", "metadata": {}, "source": [ "#### Tests" ] }, { "cell_type": "code", "execution_count": 27, "id": "b2176f6b", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "95cf8b324808d901baa863057040f86a", "grade": true, "grade_id": "subst_tests", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[36mRun starting. Expected test count is: 5\u001b[0m\n", "\u001b[32mcmd27$Helper$$anon$1:\u001b[0m\n", "\u001b[32m- should Test Case 1\u001b[0m\n", "\u001b[32m- should Test Case 2\u001b[0m\n", "\u001b[32m- should Test Case 3\u001b[0m\n", "\u001b[32m- should Test Case 4\u001b[0m\n", "\u001b[32m- should Test Case 5\u001b[0m\n", "\u001b[36mRun completed in 2 milliseconds.\u001b[0m\n", "\u001b[36mTotal number of tests run: 5\u001b[0m\n", "\u001b[36mSuites: completed 1, aborted 0\u001b[0m\n", "\u001b[36mTests: succeeded 5, failed 0, canceled 0, ignored 0, pending 0\u001b[0m\n", "\u001b[32mAll tests passed.\u001b[0m\n", "*** 🎉 Tests Passed (5 points) ***\n" ] } ], "source": [ "test(new AnyFlatSpec {\n", " val testcase1 = Var(\"A\")\n", " val testcase2 = Not(Var(\"B\"))\n", " val testcase3 = Or(Var(\"A\"), Var(\"B\"))\n", " val testcase4 = And(Not(Var(\"C\")), Or(Var(\"D\"), Var(\"E\")))\n", " val testcase5 = Or(And(Var(\"X\"), Var(\"Y\")), Or(Var(\"Z\"), Var(\"X\")))\n", "\n", " it should \"Test Case 1\" in { assertResult( Var(\"X\") ) { subst(testcase1, Var(\"X\"), \"A\") } }\n", " it should \"Test Case 2\" in { assertResult( Not(Var(\"Y\")) ) { subst(testcase2, Var(\"Y\"), \"B\") } }\n", " it should \"Test Case 3\" in { assertResult( Or(Var(\"C\"), Var(\"B\")) ) { subst(testcase3, Var(\"C\"), \"A\") } }\n", " it should \"Test Case 4\" in { assertResult( And(Not(Var(\"C\")), Or(Var(\"D\"), Var(\"F\"))) ) { subst(testcase4, Var(\"F\"), \"E\") } }\n", " it should \"Test Case 5\" in { assertResult( Or(And(Var(\"W\"), Var(\"Y\")), Or(Var(\"Z\"), Var(\"W\"))) ) { subst(testcase5, Var(\"W\"), \"X\") } }\n", "}, 5)" ] }, { "cell_type": "markdown", "id": "258e56f0-791c-487a-9bf5-44c3f5ad5f2b", "metadata": {}, "source": [ "## Concrete Syntax\n", "\n", "### Precedence Detective\n", "\n", "Consider the Scala binary expressions $$\\begin{array}{rrrl}\n", " \\text{expressions} & e& \\mathrel{::=}& n\\mid\n", " e_1 \\mathrel{\\texttt{-}} e_2 \\mid\n", " e_1 \\mathrel{\\texttt{<<}} e_2 \\\\\n", " \\text{integers} & n\n", "\\end{array}$$\n", "\n", "<span class=\"theorem-title\">**Exercise 4 (5 points)**</span> Write Scala\n", "expressions to determine if $\\texttt{-}$ has higher precedence than\n", "$\\texttt{<<}$ or vice versa. To do, write an expression bound to\n", "`e_no_parens` that uses no parentheses. Then, bind to `e_higher_-` the\n", "expression that adds parentheses to the `e_no_parens` expression\n", "corresponding to the case if $\\texttt{-}$ has higher precedence than\n", "$\\texttt{<<}$, and bind to `e_higher_<<` the expression adds parentheses\n", "corresponding to the other case.\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 28, "id": "d8f3bfb9", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "7390f29f3e77d88a7e7062e2075f457f", "grade": false, "grade_id": "detective_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[36me_no_parens\u001b[39m: \u001b[32mInt\u001b[39m = \u001b[32m160\u001b[39m\n", "\u001b[36me_higher_-\u001b[39m: \u001b[32mInt\u001b[39m = \u001b[32m160\u001b[39m\n", "\u001b[36me_higher_<<\u001b[39m: \u001b[32mInt\u001b[39m = \u001b[32m-305\u001b[39m" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "val e_no_parens = 15 - 10 << 5\n", "\n", "val e_higher_- = (15 - 10) << 5\n", "\n", "val e_higher_<< = 15 - (10 << 5)" ] }, { "cell_type": "markdown", "id": "5d01deab-3444-4389-a508-267ce2000ae1", "metadata": {}, "source": [ "Make sure that you are checking for precedence and not for left or right\n", "associativity.\n", "\n", "#### Assertions" ] }, { "cell_type": "code", "execution_count": 29, "id": "4ea4f3fa", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "aca8f5794f0463134ecc5a6431a927fc", "grade": true, "grade_id": "detective_tests", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "*** 🎉 Tests Passed (5 points) ***\n" ] } ], "source": [ "assert(e_no_parens == e_higher_- || e_no_parens == e_higher_<<)\n", "assert(e_higher_- != e_higher_<<)\n", "passed(5)" ] }, { "cell_type": "markdown", "id": "e37dcbd3-7a37-40ee-8c38-f5cf4efe04e2", "metadata": {}, "source": [ "#### Explanation\n", "\n", "<span class=\"theorem-title\">**Exercise 5 (5 points)**</span> Explain how\n", "you arrived at the relative precedence of $\\texttt{-}$ and $\\texttt{<<}$\n", "based on the output that you saw in the Scala interpreter.\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "markdown", "id": "9863319c-1d77-42fd-b2e7-736986916592", "metadata": { "deletable": false, "nbgrader": { "cell_type": "markdown", "checksum": "14b2eda93b7713aad6035f40f587a675", "grade": true, "grade_id": "detective_explanation", "locked": false, "points": 5, "schema_version": 3, "solution": true, "task": false } }, "source": [ "1. `e_no_parens`: When written without parentheses, it is up to Scala interpreter to determine precedence between subtraction and left shift.\n", "2. `e_higher_-`: This expresion will have parentheses to explicitly show that subtraction has higher precedence than left shift.\n", "3. `e_higher_<<`: This expression will have parentheses to explicitly show that left shift has high precedence than subtraction." ] }, { "cell_type": "markdown", "id": "6b00d249-372d-44df-9c44-aecd94bce0a4", "metadata": {}, "source": [ "## Parse Trees\n", "\n", "Consider the following grammar:\n", "\n", "<span id=\"eq-ite-grammar\">$$\\begin{array}{rrrl}\n", " \\text{expressions} & e& \\mathrel{::=}& x\\mid e\\;\\texttt{?}\\;e\\;\\texttt{:}\\;e \\\\\n", " \\text{variables} & x& \\mathrel{::=}& \\texttt{a} \\mid\\texttt{b}\n", "\\end{array} \\qquad(1)$$</span>\n", "\n", "Consider the following data type for representing parse trees:" ] }, { "cell_type": "code", "execution_count": 30, "id": "e0ff9277", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "0cda8ee79fbaceb83e15d3e0117e877c", "grade": false, "grade_id": "parsetree_types", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mtrait\u001b[39m \u001b[36mParseTree\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mLeaf\u001b[39m\n", "defined \u001b[32mclass\u001b[39m \u001b[36mNode\u001b[39m" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sealed trait ParseTree\n", "case class Leaf(term: String) extends ParseTree\n", "case class Node(nonterm: String, children: List[ParseTree]) extends ParseTree" ] }, { "cell_type": "markdown", "id": "9b5b444d-6085-494d-8015-cd9f56a46003", "metadata": {}, "source": [ "Observe that a `ParseTree` is just an $n$-ary tree containing `String`s.\n", "A `Leaf` is a terminal containing the lexemes (i.e., letters from the\n", "alphabet), while a `Node` is represents a non-terminal with a `String`\n", "for the name of the non-terminal a list of `children` `ParseTree`s.\n", "\n", "For example, the following are `ParseTree`s for this grammar\n", "(<a href=\"#eq-ite-grammar\" class=\"quarto-xref\">Equation 1</a>):" ] }, { "cell_type": "code", "execution_count": 31, "id": "aebbb3d1", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "bdee80feebaa951f42684815b22c642d", "grade": false, "grade_id": "parsetree_examples", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[36mp1\u001b[39m: \u001b[32mNode\u001b[39m = \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m)))\n", "\u001b[36mp2\u001b[39m: \u001b[32mNode\u001b[39m = \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m))))\n", ")" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "val p1 = Node(\"x\", Leaf(\"a\") :: Nil)\n", "val p2 = Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)" ] }, { "cell_type": "markdown", "id": "0df2099c-8f51-4cb1-9ce3-77895f8c7d24", "metadata": {}, "source": [ "We provide a function that pretty-prints a `ParseTree` for *this\n", "grammar* (<a href=\"#eq-ite-grammar\" class=\"quarto-xref\">Equation 1</a>)." ] }, { "cell_type": "code", "execution_count": 32, "id": "9c07f06e", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "d45ba978ac101ae4f1409cf49329b416", "grade": false, "grade_id": "parsetree_pretty", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mfunction\u001b[39m \u001b[36mpretty\u001b[39m\n", "\u001b[36mres32_1\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[33mSome\u001b[39m(value = \u001b[32m\"a\"\u001b[39m)\n", "\u001b[36mres32_2\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[33mSome\u001b[39m(value = \u001b[32m\"a\"\u001b[39m)" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def pretty(t: ParseTree): Option[String] = {\n", " val alphabet = Set(\"a\", \"b\", \"?\", \":\")\n", " t match {\n", " // e ::= x\n", " case Node(\"e\", (x @ Node(\"x\", _)) :: Nil) => pretty(x)\n", " // e ::= e ? e : e\n", " case Node(\"e\", (e1 @ Node(\"e\", _)) :: Leaf(\"?\") :: (e2 @ Node(\"e\", _)) :: Leaf(\":\") :: (e3 @ Node(\"e\", _)) :: Nil) =>\n", " for { s1 <- pretty(e1); s2 <- pretty(e2); s3 <- pretty(e3) }\n", " yield s\"$s1 ? $s2 : $s3\"\n", " // x ::= a\n", " case Node(\"x\", Leaf(\"a\") :: Nil) => Some(\"a\")\n", " // x ::= b\n", " case Node(\"x\", Leaf(\"b\") :: Nil) => Some(\"b\")\n", " // failure\n", " case _ => None\n", " }\n", "}\n", "\n", "pretty(p1)\n", "pretty(p2)" ] }, { "cell_type": "markdown", "id": "825bfb1e-0618-42ff-8ce7-8a99ffafbe82", "metadata": {}, "source": [ "Since it is possible to have `ParseTree`s that are not recognized by\n", "this grammar, the `pretty` function has return type `Option[String]`.\n", "Calling `pretty` on `ParseTree`s that are not in this grammar return\n", "`None`:" ] }, { "cell_type": "code", "execution_count": 33, "id": "f5778462", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "4e383bc90701275803c2d59457e1ffdb", "grade": false, "grade_id": "parsetree_pretty_none", "locked": true, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[36mres33_0\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[32mNone\u001b[39m\n", "\u001b[36mres33_1\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[32mNone\u001b[39m\n", "\u001b[36mres33_2\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[32mNone\u001b[39m" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pretty(Leaf(\"a\"))\n", "pretty(Node(\"x\", Leaf(\"c\") :: Nil))\n", "pretty(Node(\"y\", Leaf(\"a\") :: Nil))" ] }, { "cell_type": "markdown", "id": "f487d822-bbb6-48c4-bdd8-510cb0fbe6a3", "metadata": {}, "source": [ "Note that the `pretty` function makes use of Scala’s `for`-`yield`\n", "expressions, which you do not need to understand for this exercise.\n", "\n", "#### Exercise\n", "\n", "<span class=\"theorem-title\">**Exercise 6 (5 points)**</span> Give a\n", "parse tree for the sentence in the grammar `a ? b : a`.\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 34, "id": "e7d3515d", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "f4a6165e9338422805ad7a5697a15209", "grade": false, "grade_id": "parse_tree_1_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[36mparseTree1\u001b[39m: \u001b[32mParseTree\u001b[39m = \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m))))\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\"?\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"b\"\u001b[39m))))\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\":\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m))))\n", " )\n", " )\n", ")" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "val parseTree1: ParseTree =\n", " Node(\"e\",\n", " Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Leaf(\"?\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"b\") :: Nil) :: Nil)\n", " :: Leaf(\":\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Nil\n", " )" ] }, { "cell_type": "markdown", "id": "32bcc28b-473e-4af8-b5a4-125b3e98718c", "metadata": {}, "source": [ "#### Assertion" ] }, { "cell_type": "code", "execution_count": 35, "id": "60cc7566", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "235a3c106ebba2dd23cc05c34da836bf", "grade": true, "grade_id": "parsetree_tree_test", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "*** 🎉 Tests Passed (5 points) ***\n" ] } ], "source": [ "assert(pretty(parseTree1) == Some(\"a ? b : a\"))\n", "passed(5)" ] }, { "cell_type": "markdown", "id": "4127541f-87d2-49d5-8bae-045e20feac58", "metadata": {}, "source": [ "#### Exercise\n", "\n", "<span class=\"theorem-title\">**Exercise 7 (10 points)**</span> Show that\n", "this grammar\n", "(<a href=\"#eq-ite-grammar\" class=\"quarto-xref\">Equation 1</a>) is\n", "ambiguous by giving two parse trees for the sentence in the grammar\n", "`a ? b : a ? b : a`.\n", "\n", "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 36, "id": "e0c46f00", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "fec8c17457a1f15c9d4eb305bc5193b9", "grade": false, "grade_id": "parse_tree_ambiguous_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "\u001b[36mparseTree2\u001b[39m: \u001b[32mParseTree\u001b[39m = \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m))))\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\"?\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"b\"\u001b[39m))))\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\":\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m)))\n", " )\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\"?\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"b\"\u001b[39m)))\n", " )\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\":\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m)))\n", " )\n", " )\n", " )\n", " )\n", " )\n", ")\n", "\u001b[36mparseTree3\u001b[39m: \u001b[32mParseTree\u001b[39m = \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m)))\n", " )\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\"?\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"b\"\u001b[39m)))\n", " )\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\":\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\n", " \u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m)))\n", " )\n", " )\n", " )\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\"?\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"b\"\u001b[39m))))\n", " ),\n", " \u001b[33mLeaf\u001b[39m(term = \u001b[32m\":\"\u001b[39m),\n", " \u001b[33mNode\u001b[39m(\n", " nonterm = \u001b[32m\"e\"\u001b[39m,\n", " children = \u001b[33mList\u001b[39m(\u001b[33mNode\u001b[39m(nonterm = \u001b[32m\"x\"\u001b[39m, children = \u001b[33mList\u001b[39m(\u001b[33mLeaf\u001b[39m(term = \u001b[32m\"a\"\u001b[39m))))\n", " )\n", " )\n", ")" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "// a ? b : (a ? b : a)\n", "val parseTree2: ParseTree =\n", " Node(\"e\",\n", " Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Leaf(\"?\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"b\") :: Nil) :: Nil)\n", " :: Leaf(\":\")\n", " :: Node(\"e\",\n", " Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Leaf(\"?\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"b\") :: Nil) :: Nil)\n", " :: Leaf(\":\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Nil)\n", " :: Nil\n", " )\n", "\n", "// (a ? b : a) ? b : a\n", "val parseTree3: ParseTree =\n", " Node(\"e\",\n", " Node(\"e\",\n", " Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Leaf(\"?\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"b\") :: Nil) :: Nil)\n", " :: Leaf(\":\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Nil)\n", " :: Leaf(\"?\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"b\") :: Nil) :: Nil)\n", " :: Leaf(\":\")\n", " :: Node(\"e\", Node(\"x\", Leaf(\"a\") :: Nil) :: Nil)\n", " :: Nil\n", " )" ] }, { "cell_type": "code", "execution_count": 37, "id": "789ba753", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[36mres37_0\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[33mSome\u001b[39m(value = \u001b[32m\"a ? b : a ? b : a\"\u001b[39m)\n", "\u001b[36mres37_1\u001b[39m: \u001b[32mOption\u001b[39m[\u001b[32mString\u001b[39m] = \u001b[33mSome\u001b[39m(value = \u001b[32m\"a ? b : a ? b : a\"\u001b[39m)" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pretty(parseTree2)\n", "pretty(parseTree3)" ] }, { "cell_type": "markdown", "id": "b6096f88-0de5-4826-9599-d9c86f2b365e", "metadata": {}, "source": [ "#### Assertions" ] }, { "cell_type": "code", "execution_count": 38, "id": "72197a91", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "1072c2352550ba63657d754309a8a368", "grade": true, "grade_id": "parsetree_ambiguous_test_1", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "*** 🎉 Tests Passed (5 points) ***\n" ] } ], "source": [ "assert(parseTree2 != parseTree3)\n", "assert(pretty(parseTree2).isDefined)\n", "assert(pretty(parseTree3).isDefined)\n", "passed(5)" ] }, { "cell_type": "code", "execution_count": 39, "id": "084fbb3b", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "1843616ac2befc8ef5c950df69e798aa", "grade": true, "grade_id": "parsetree_ambiguous_test_2", "locked": true, "points": 5, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "*** 🎉 Tests Passed (5 points) ***\n" ] } ], "source": [ "assert(pretty(parseTree2) == Some(\"a ? b : a ? b : a\"))\n", "assert(pretty(parseTree3) == Some(\"a ? b : a ? b : a\"))\n", "passed(5)" ] }, { "cell_type": "markdown", "id": "558f8a4c-7af1-4eaa-bd10-99dae585dc75", "metadata": {}, "source": [ "## Defining Grammars\n", "\n", "In this question, we define a BNF grammar for *floating-point numbers*\n", "that are made up of a fraction (e.g., `5.6` or `3.123` or `-2.5`)\n", "followed by an optional exponent (e.g., `E10` or `E-10`).\n", "\n", "More precisely for this exercise, our floating-point numbers\n", "\n", "- [x] must have a decimal point,\n", "- [x] do not have leading zeros,\n", "- [x] can have any number of trailing zeros,\n", "- [x] non-zero exponents if it exists,\n", "- [x] must have non-zero fraction to have an exponent, and\n", "- [x] cannot have a ‘-’ in front of a zero number; also,\n", "- [x] the exponent cannot have leading zeros.\n", "\n", "The exponent, if it exists, is the letter `E` followed by an integer.\n", "For example, the following are floating-point numbers: `0.0`, `3.5E3`,\n", "`3.123E30`, `-2.5E2`, `-2.5E-2`, `3.50`, and `3.01E2`.\n", "\n", "The following are examples of strings that are *not* floating-point\n", "numbers by our definition: `0`, `-0.0`, `3.E3`, `E3`, `3.0E4.5`, and\n", "`4E4`.\n", "\n", "For this exercise, let us assume that the tokens are characters in the\n", "following alphabet $\\Sigma$:\n", "\n", "$\\Sigma \\stackrel{\\text{\\tiny def}}{=}\\{$ `0`, `1`, `2`, `3`, `4`, `5`,\n", "`6`, `7`, `8`, `9`, `E`, `-`, `.` $\\}$\n", "\n", "<span class=\"theorem-title\">**Exercise 8 (10 points)**</span> Translate\n", "a grammar into a parser `FloatParser.float: Parser[String]` using the\n", "Scala Parsing Combinator Library that recognizes the language of\n", "floating-point numbers described above. Since we do not care about the\n", "correctness of the grammar, we let parse result simply be the input\n", "string.\n", "\n", "We suggest that you first use the scratch cell below to give a grammar\n", "in BNF notation. Your grammar should be completely defined using the\n", "alphabet above as the terminals (i.e., it should not count on a\n", "non-terminal that it does not itself define)." ] }, { "cell_type": "markdown", "id": "8c1ba46e-527e-4664-a32f-80e800bbfe7f", "metadata": {}, "source": [ "SCRATCH CELL" ] }, { "cell_type": "markdown", "id": "a15d3aab-2045-4565-87bd-46ce02998967", "metadata": {}, "source": [ "**Edit this cell:**" ] }, { "cell_type": "code", "execution_count": 40, "id": "1355139a", "metadata": { "deletable": false, "nbgrader": { "cell_type": "code", "checksum": "2c18dc9a164093ed57559bc0bee3e557", "grade": false, "grade_id": "floatparser_answer", "locked": false, "schema_version": 3, "solution": true, "task": false } }, "outputs": [ { "data": { "text/plain": [ "defined \u001b[32mobject\u001b[39m \u001b[36mFloatParser\u001b[39m" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "object FloatParser extends scala.util.parsing.combinator.RegexParsers {\n", " val concat2: String ~ String => String = { case s1 ~ s2 => s1 + s2 }\n", " val concat3: String ~ String ~ String => String = { case s1 ~ s2 ~ s3 => s1 + s2 + s3 }\n", " val concat4: String ~ String ~ String ~ String => String = { case s1 ~ s2 ~ s3 ~ s4 => s1 + s2 + s3 + s4 }\n", " val concat5: String ~ String ~ String ~ String ~ String => String = { case s1 ~ s2 ~ s3 ~ s4 ~ s5 => s1 + s2 + s3 + s4 + s5 }\n", "\n", " def float: Parser[String] =\n", " fraction ~ exponent ^^ concat2 |\n", " sign ~ fraction ~ exponent ^^ concat3 |\n", " fraction |\n", " success(\"\")\n", "\n", " def fraction: Parser[String] =\n", " anyOneToNine ~ \".\" ~ oneOrMoreDigits ^^ concat3 |\n", " \"0\" ~ \".\" ~ zeroOrMoreDigits ^^ concat3\n", "\n", " def oneOrMoreDigits: Parser[String] =\n", " digit ~ zeroOrMoreDigits ^^ concat2\n", "\n", " def exponent: Parser[String] =\n", " \"E\" ~ sign ~ anyOneToNine ~ zeroOrMoreDigits ^^ concat4 |\n", " \"E\" ~ anyOneToNine ~ zeroOrMoreDigits ^^ concat3\n", " success(\"\")\n", "\n", " def zeroOrMoreDigits: Parser[String] =\n", " digit ~ zeroOrMoreDigits ^^ concat2 |\n", " success(\"\")\n", "\n", " def digit: Parser[String] = \"0\" | anyOneToNine\n", "\n", " def anyOneToNine: Parser[String] = \"1\" | \"2\" | \"3\" | \"4\" | \"5\" | \"6\" | \"7\" | \"8\" | \"9\"\n", "\n", " def sign: Parser[String] =\n", " \"-\" |\n", " success(\"\")\n", "\n", " def parse(str: String): Option[String] = parseAll(float, str) match {\n", " case Success(str, _) => Some(str)\n", " case Failure(_, _) | Error(_, _) => None\n", " }\n", "}" ] }, { "cell_type": "markdown", "id": "327ebd79-bbe0-4763-a6c4-b71a860f3a3a", "metadata": {}, "source": [ "#### Notes\n", "\n", "- The `success[A](a: A): Parser[A]` method corresponds to an\n", " $\\varepsilon$ production in BNF. It returns a parser that is\n", " successful without consuming any input yielding the parse result\n", " `a`. If you use it in your parser, make sure it is the *last*\n", " production for the non-terminal.\n", "- We provide some helper functions\n", " `concat2: String ~ String => String`,\n", " `concat3: String ~ String ~ String => String`, etc. that simply\n", " concatenate their input strings together. These helper functions are\n", " the only semantic actions you need.\n", "- We have provided some basic parsers `sign`, `anyOneToNine`, `digit`,\n", " `zeroOrMoreDigits` that you may use if you like.\n", "- You may edit the `parse` interface function that we have provided to\n", " start from a different non-terminal while you’re developing, but\n", " make sure it is `float` in the end and that your starting\n", " non-terminal is `float`. Alternatively, you may add additional such\n", " interface functions with a different name for your testing.\n", "\n", "#### Tests" ] }, { "cell_type": "code", "execution_count": 41, "id": "b1824177", "metadata": { "deletable": false, "editable": false, "nbgrader": { "cell_type": "code", "checksum": "66ebc679977f9e9d6d9450c594637910", "grade": true, "grade_id": "floatparser_tests", "locked": true, "points": 10, "schema_version": 3, "solution": false, "task": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[36mRun starting. Expected test count is: 13\u001b[0m\n", "\u001b[32mcmd41$Helper$$anon$1:\u001b[0m\n", "\u001b[32mFloatParser\u001b[0m\n", "\u001b[32m- should parse 0.0\u001b[0m\n", "\u001b[32m- should parse 3.5E3\u001b[0m\n", "\u001b[32m- should parse 3.123E30\u001b[0m\n", "\u001b[32m- should parse -2.5E2\u001b[0m\n", "\u001b[32m- should parse -2.5E-2\u001b[0m\n", "\u001b[32m- should parse 3.50\u001b[0m\n", "\u001b[32m- should parse 3.01E2\u001b[0m\n", "\u001b[32m- should fail to parse 0\u001b[0m\n", "\u001b[32m- should fail to parse -0.0\u001b[0m\n", "\u001b[32m- should fail to parse 3.E3\u001b[0m\n", "\u001b[32m- should fail to parse E3\u001b[0m\n", "\u001b[32m- should fail to parse 3.0E4.5\u001b[0m\n", "\u001b[32m- should fail to parse 4E4\u001b[0m\n", "\u001b[36mRun completed in 15 milliseconds.\u001b[0m\n", "\u001b[36mTotal number of tests run: 13\u001b[0m\n", "\u001b[36mSuites: completed 1, aborted 0\u001b[0m\n", "\u001b[36mTests: succeeded 13, failed 0, canceled 0, ignored 0, pending 0\u001b[0m\n", "\u001b[32mAll tests passed.\u001b[0m\n", "*** 🎉 Tests Passed (10 points) ***\n" ] } ], "source": [ "test(new AnyFlatSpec {\n", " val tests_positive = List(\n", " \"0.0\",\n", " \"3.5E3\",\n", " \"3.123E30\",\n", " \"-2.5E2\",\n", " \"-2.5E-2\",\n", " \"3.50\",\n", " \"3.01E2\"\n", " )\n", "\n", " behavior of \"FloatParser\"\n", "\n", " for (t <- tests_positive) {\n", " it should s\"parse $t\" in {\n", " assertResult( Some(t) ) {\n", " FloatParser.parse(t)\n", " }\n", " }\n", " }\n", "\n", " val tests_negative = List(\n", " \"0\",\n", " \"-0.0\",\n", " \"3.E3\",\n", " \"E3\",\n", " \"3.0E4.5\",\n", " \"4E4\"\n", " )\n", "\n", " for (t <- tests_negative) {\n", " it should s\"fail to parse $t\" in {\n", " assertResult( None ) {\n", " FloatParser.parse(t)\n", " }\n", " }\n", " }\n", "}, 10)" ] } ], "metadata": { "kernelspec": { "display_name": "Scala", "language": "scala", "name": "scala" }, "language_info": { "codemirror_mode": "text/x-scala", "file_extension": ".sc", "mimetype": "text/x-scala", "name": "scala", "nbconvert_exporter": "script", "version": "2.13.14" } }, "nbformat": 4, "nbformat_minor": 5 } |