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Symbolic unit checking

A symbolic unit checker examines an equation before any numerical values are inserted. It assigns dimensions to symbols, propagates them through an expression tree, and requires both sides of an equality to match. The checker is most useful when it reports the first operation that became invalid rather than returning only false.

Declare a symbol table

Before checking an equation, declare the dimensions of its symbols. For one-dimensional constant-acceleration motion,

\[x=x_0+v_0t+\frac{1}{2}at^2\]

the relevant declarations are

\[[x]=[x_0]=L,\qquad[v_0]=LT^{-1},\qquad[t]=T,\qquad[a]=LT^{-2}\]

The numerical factor \(1/2\) is dimensionless. Every symbol must be declared. Treating an unknown symbol as dimensionless can make an invalid expression appear valid.

from dataclasses import dataclass


@dataclass(frozen=True)
class Dimension:
    mass: int
    length: int
    time: int


SymbolTable = dict[str, Dimension]

Production code may include all seven SI base dimensions and rational rather than floating-point exponents. The smaller structure keeps the algorithm visible.

Parse before checking

The source x0 + v0 * t must become a tree that preserves grouping and operator precedence. Manipulating the source with regular-expression replacements is fragile: a symbol name may contain another name, and powers or parentheses change structure.

from dataclasses import dataclass
from typing import Literal


@dataclass(frozen=True)
class Expression:
    kind: Literal["number", "symbol", "binary", "power"]
    location: int
    value: float | None = None
    name: str | None = None
    operator: Literal["+", "-", "*", "/"] | None = None
    left: "Expression | None" = None
    right: "Expression | None" = None
    base: "Expression | None" = None
    exponent: int | None = None

Locations allow an error to point to the relevant operator. A parser should reject malformed syntax before dimension evaluation begins.

Evaluate dimension rules

A numerical literal is dimensionless. A symbol returns its declared vector. Multiplication adds vector exponents and division subtracts them. A fixed power multiplies every exponent by the power.

Addition and subtraction are different: their operands must already have equal dimensions, and the result retains that dimension.

def dimension_of(node, symbols):
    if node.kind == "number":
        return Dimension(0, 0, 0)
    if node.kind == "symbol":
        return symbols[node.name]
    if node.kind == "power":
        base = dimension_of(node.base, symbols)
        return Dimension(
            base.mass * node.exponent,
            base.length * node.exponent,
            base.time * node.exponent,
        )
    left = dimension_of(node.left, symbols)
    right = dimension_of(node.right, symbols)
    if node.operator in {"+", "-"}:
        if left != right:
            raise ValueError(f"Incompatible units at {node.location}.")
        return left
    sign = 1 if node.operator == "*" else -1
    return Dimension(
        left.mass + sign * right.mass,
        left.length + sign * right.length,
        left.time + sign * right.time,
    )


symbols = {"v": Dimension(0, 1, -1), "t": Dimension(0, 0, 1)}
product = Expression(
    kind="binary",
    location=1,
    operator="*",
    left=Expression(kind="symbol", location=0, name="v"),
    right=Expression(kind="symbol", location=2, name="t"),
)
assert dimension_of(product, symbols) == Dimension(0, 1, 0)
print("Velocity times time has length dimensions.")

After evaluating both sides of an equation, compare their vectors. The equality is dimensionally consistent only when all corresponding exponents match.

Walk through a valid equation

In the constant-acceleration equation, the velocity term reduces to

\[[v_0t]=(LT^{-1})T=L\]

and the acceleration term reduces to

\[[at^2]=(LT^{-2})T^2=L\]

The initial position, velocity term, and acceleration term are all lengths, so their addition is allowed. The right side reduces to length, matching the left side.

Now replace the final power with a cube:

\[x=x_0+v_0t+\frac{1}{2}at^3\]

The last term has dimension \(LT\), so the checker should flag the addition of that term to a length. Reporting only that the whole equation fails would hide the useful location.

Support functions deliberately

Trigonometric, exponential, and logarithmic functions require dimensionless arguments. For

\[x(t)=A\cos(\omega t+\phi)\]

the phase must be dimensionless. The checker therefore verifies that angular frequency has inverse-time dimension and that \(\omega t\) can be added to dimensionless phase \(\phi\). The cosine result is dimensionless, so multiplication by amplitude gives the dimension of position.

Square root is a power of one half. Absolute value preserves dimension. Functions such as min and max require compatible arguments. A generic function call should not be assumed dimensionless; each supported function needs an explicit rule, and unknown functions should fail or remain symbolic.

Angles deserve an extra semantic tag even though radians are dimensionless in SI. This can prevent a plain ratio or strain from being passed to a trigonometric function by accident.

Derivatives and integrals change dimensions

Symbolic physics often includes calculus. Differentiating with respect to time divides the result dimension by time. If position has length dimension, then

\[\left[\frac{\mathrm{d}x}{\mathrm{d}t}\right]=LT^{-1}\]

and a second time derivative has acceleration dimension. Integrating with respect to time multiplies by time. Definite integration limits must have the same dimension as the integration variable.

This allows a checker to catch a missing differential or a derivative taken with respect to the wrong kind of quantity. It still cannot verify differentiability or the correctness of an antiderivative.

Unit scale is a second pass

Dimension checking says that centimetres and metres are compatible lengths. It does not perform their scale conversion. An evaluator that also handles units needs a canonical representation and conversion graph. Addition may then convert compatible operands to one unit before combining magnitudes.

Affine units such as degrees Celsius require an offset for absolute temperatures, while temperature differences use only a scale. Treating every conversion as multiplication would be wrong. Symbolic dimensions should remain separate from these unit-specific rules.

Tests for the checker

Build tests from algebraic identities and known failures:

Property-based tests can generate exponent vectors and verify inverse operations, such as multiplying then dividing by the same nonzero-dimensional quantity. Parser tests and dimension tests should remain separate so a syntax defect is not misdiagnosed as a physics error.

What the checker cannot prove

Dimensional consistency is necessary for a valid physical equation, but not sufficient. The expressions \(K=mv^2\) and \(K=\tfrac12mv^2\) share dimensions, but only the second gives classical translational kinetic energy. Dimension analysis cannot determine dimensionless constants or signs.

It also cannot distinguish all semantic quantities with the same dimensions. Torque and energy both reduce to force times length. Two positions in different coordinate frames are both lengths. A field value can have correct dimensions while being evaluated at the wrong location.

Use the checker as an early gate that catches a broad class of mistakes cheaply. Follow it with source comparison, algebraic review, limiting cases, numerical tests, and physical validation. The checker should make review more focused, not replace it. Clear diagnostics also teach the dimensional reasoning that produced each result.

A worked Python notebook

A 2 kilogram body starts at 3 metres per second and accelerates at 1 metre per second squared. The question for this calculation is precise: how does kinetic energy change when we change duration, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The units of the velocity sum must be checked before squaring it. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 4 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for duration, expressed in s. The output is kinetic energy, expressed in J. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \left(q + 3\right)^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 2 s, the expression gives 25 J. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("2*(3+q)**2/2", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted kinetic energy:", reference_output, "J")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 9 J. The midpoint, at 2 s, gives 25 J. The upper endpoint gives 49 J. Read these together with the assumptions: they describe different choices of duration within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 2.0, 4.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} J")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows duration; the vertical axis shows kinetic energy. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 4.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Duration (s)")
ax.set_ylabel("Kinetic energy (J)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does kinetic energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is 10 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 2 q + 6\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 2.02
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 2.02 s. The tangent estimate is 25.2 J, while the full expression gives 25.2004 J. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 4.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of duration inside the stated interval, predict whether kinetic energy will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The units of the velocity sum must be checked before squaring it. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of J: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(4.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 4.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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