Now let\u2019s define the term structurally equal<\/em> to mean that the formal way the symbols are written in the expression are the same even if the identity of the symbols are not. For example,<\/p>\n\n\n\n \\[ D\u2019\u2019 = \\sqrt{ (q-q_0)^2 + p^2 } \\; \\]<\/p>\n\n\n\n is structurally equal to the expression for $D$ since we recognize that the symbol substitutions<\/p>\n\n\n\n \\[ x \\rightarrow q \\; ,\\]<\/p>\n\n\n\n \\[ a \\rightarrow q_0 \\; , \\]<\/p>\n\n\n\n and<\/p>\n\n\n\n \\[ y \\rightarrow p \\; \\]<\/p>\n\n\n\n make $D$ look the same on paper as $D\u2019\u2019$. Note that two expressions that are structurally equal need not be mathematically equal if assumptions about the different symbols aren\u2019t the same. For example, if $x \\in (-\\infty,\\infty)$ but we restrict $p \\in [0,\\infty]$, then, despite their structurally equality $D$ is not mathematically equal to $D\u2019\u2019$ when $x < 0$.<\/p>\n\n\n\n We will use the term exactly equal<\/em> to mean that two expressions are both mathematical equal and structurally equal and have the same symbols.<\/p>\n\n\n\n These three definitions have holes and limitations. The holes are a by-product of limitations of human logic and we won\u2019t try to patch them so much as work around them when the time comes. Regarding the limitations, we can give a general notion of where they will show up and then revisit them in the future. The primary limitation(s) is that the notion of equivalency is left out. To give a flavor of this consider the two expressions<\/p>\n\n\n\n \\[ \\frac{d}{dx} \\left( x^2 – 3 a x + 9 \\right) \\; \\]<\/p>\n\n\n\n and<\/p>\n\n\n\n \\[ 2x -3a \\; .\\]<\/p>\n\n\n\n These two expressions are neither mathematically equal (one can\u2019t simply substitute in a value for $x$ before taking the derivative) nor structurally equal (the symbol structure isn\u2019t the same). But they are equivalent in the sense that applying the derivative in the first leads one to the second. And, there is another wrinkle when considering moving from the second expression to the first, in that $2x – 3a$ is equivalent to an infinite number of expressions of the form<\/p>\n\n\n\n \\[ \\frac{d}{dx} \\left( x^2 – 3 a x + constant \\right) \\; .\\]<\/p>\n\n\n\n Since we will have our hand full just dealing with how to teach an agent how to determine if $D$ is structurally or mathematically equal to $D\u2019$, we will defer these deeper matters and look at a simple example from basic physics.<\/p>\n\n\n\n It is typical for a professor, when teaching say electromagnetism, to look at $D\u2019$ and simply highlight the first three terms under the radical and say something to the effect that the form a perfect square which can be \u2018reduced\u2019 or \u2018simplified\u2019 to the other.<\/p>\n\n\n However, there is no cognitive mind behind a computer (no matter how much training data it may have ingested) and so it can\u2019t fill in the gaps and move (albeit not usually effortlessly) between the various ambiguities and elastic meanings in the way a human can. <\/p>\n\n\n\n To understand this point better, consider that to represent the expression above requires nesting $x^2 – 2 a x + a^2 + y^2$ under a square root symbol. That\u2019s four terms \u2018owned\u2019 by the square root, which we wish to \u2018factor\u2019 into two terms $(x-a)^2 + y^2$. In addition, each of these terms is complicated as none are \u2018atomic\u2019. A term is atomic if it consists of a symbol and nothing else.<\/p>\n\n\n\n Driving this point home is easier done with a visual. Using the graphviz application and the corresponding Python API, we can visually display how these various expressions are represented internally. SymPy uses a tree structure that, for the expression $D\u2019$, looks like<\/p>\n\n\n Every node in the a SymPy tree is either a function or a symbol. Functions own (almost always) children nodes reflecting their composite. Symbols are terminal nodes reflecting their atomic nature. At the top of the tree is the Pow function (for power) with two main branches: Add and Half. Add is the function that owns the four terms that algebraically add together while Half is a special symbol meaning 1\/2. SymPy reserves a special symbol for this since division by 2 is so common. Of the four main branches of Add, three are Pow and one is Mul (for multiply). Like Add, Mul can own an arbitrary number of branches. In this case there are three, each terminating with the symbols $-2$, $a$, and $x$. <\/p>\n\n\n\n In order to manipulate the only some of the contents under the square root we must be able to find that portion of the tree that corresponds to $x^2 \u2013 2 a x + a^2$, remove it, manipulate it, and then return the new structure to the tree so that it looks like:<\/p>\n\n\n Getting the contents of the square root is relatively simple: we simple ask for the arguments of the expression and we get a tuple containing the Add branch and the Half Symbol.<\/p>\n\n\n\n The Add branch is now a polynomial expression that we might be tempted to try SymPy\u2019s factor on. However, Factor doesn\u2019t know what to do with the portion involving $y^2$. However, if we isolate the portion of the expression involving just $x$ and $a$ by subtracting off the $y^2$ piece, factoring, and then adding $y^2$ back, we get a reasonable result. Both of these approaches are shone in the notebook snippet below:<\/p>\n\n\n![\"\"](\"http:\/\/aristotle2digital.blogwyrm.com\/wp-content\/uploads\/2026\/01\/A2D_01Jan_2026_sqrt_simplification.png\")
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