From b3f6fd14aabd748f808f8a5f106297e91564b8a6 Mon Sep 17 00:00:00 2001
From: Andreas Zeller <zeller@cispa.de>
Date: Mon, 6 Jan 2025 15:40:02 +0100
Subject: [PATCH] Fix: no leading zeros in `CALC_GRAMMAR`

---
 notebooks/Alhazen.ipynb | 41 ++++++++++++++++-------------------------
 1 file changed, 16 insertions(+), 25 deletions(-)

diff --git a/notebooks/Alhazen.ipynb b/notebooks/Alhazen.ipynb
index 73644f28..5d981ba3 100644
--- a/notebooks/Alhazen.ipynb
+++ b/notebooks/Alhazen.ipynb
@@ -138,13 +138,6 @@
     "Additional test inputs are generated and executed to refine or refute the hypothesis, eventually obtaining a prediction model of the circumstances of why the behavior in question takes place."
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<!-- _Alhazen_ is a tool that automatically learns the circumstances of program failure by associating syntactical features of sample inputs with the execution outcome. The produced explanations (in the form of a decision tree) help developers focus on the input space's relevant aspects. -->"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -167,11 +160,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In a nutshell, this is how Alhazen works.\n",
+    "Let us give a high-level description of how Alhazen works, illustrated above.\n",
+    "\n",
     "Alhazen is given an _input grammar_ and a number of _input files_ (whose format is given by the grammar),\n",
     "and produces a _decision tree_ – a machine learning model that explains under which circumstances the program fails.\n",
     "\n",
-    "Alhazen determines and refines these decision trees in five steps, illustrated above.\n",
+    "Alhazen determines and refines these decision trees in five steps:\n",
     "\n",
     "1. For each input file, Alhazen extracts a number of _input features_ that apply.\n",
     "   These input features are predicates over the individual elements of the input grammar, such as `<expr> > 0` (an `<expr>` element is larger than zero) or `exists(<minus-sign>)` (the input contains a minus sign).\n",
@@ -277,14 +271,20 @@
     "    \"<term>\": [\"-<value>\", \"<value>\"],\n",
     "\n",
     "    \"<value>\":\n",
-    "        [\"<integer>.<integer>\",\n",
+    "        [\"<integer>.<digits>\",\n",
     "         \"<integer>\"],\n",
     "\n",
     "    \"<integer>\":\n",
-    "        [\"<digit><integer>\", \"<digit>\"],\n",
+    "        [\"<lead-digit><digits>\", \"<digit>\"],\n",
+    "\n",
+    "    \"<digits>\":\n",
+    "        [\"<digit><digits>\", \"<digit>\"],\n",
+    "\n",
+    "    \"<lead-digit>\":  # First digit cannot be zero\n",
+    "        [\"1\", \"2\", \"3\", \"4\", \"5\", \"6\", \"7\", \"8\", \"9\"],\n",
     "\n",
     "    \"<digit>\":\n",
-    "        [\"0\", \"1\", \"2\", \"3\", \"4\", \"5\", \"6\", \"7\", \"8\", \"9\"]\n",
+    "        [\"0\", \"1\", \"2\", \"3\", \"4\", \"5\", \"6\", \"7\", \"8\", \"9\"],\n",
     "}"
    ]
   },
@@ -292,14 +292,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We see that the `CALCULATOR` Grammar consists of several production rules. The calculator subject will only accept inputs that conform to this grammar definition."
+    "We see that the `CALC_GRAMMAR` consists of several production rules. The calculator subject will only accept inputs that conform to this grammar definition."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Now, lets load two initial input samples:"
+    "Now, let us load two initial input samples:\n",
+    "- `sqrt(-16)`\n",
+    "- `sqrt(4)`"
    ]
   },
   {
@@ -312,17 +314,6 @@
     "initial_sample_list = ['sqrt(-16)', 'sqrt(4)']"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The two initial input samples for our calculator should be:\n",
-    "- _sqrt(-16)_\n",
-    "- _sqrt(4)_\n",
-    "\n",
-    "Let's check if this is true with python's `assert` function. The condition is True, if no Assertion is thrown."
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},