Patternmatching mathematical expressions¶
Warning
The patternmatching algorithm is new and experimental. It opens up many new possibilities for providing adaptive feedback and manipulating mathematical expressions in general, but the interface is currently quite cumbersome. This documentation will be expanded as the system is used more.
Numbas includes a sophisticated patternmatching algorithm for mathematical expressions. It’s mainly used by the \simplify
command, but can also be used in, for example, custom marking scripts to answer more nuanced questions about the form of the student’s answer.
The patternmatcher should be considered to work similarly to how a regular expression algorithm, except it operates on algebraic syntax trees instead of text strings. The algorithm will either return false
, when the expression doesn’t match the pattern, or true
, and a dictionary of named matching groups, which are subtrees of the input expression.
Using the pattern matcher¶
The function Numbas.jme.display.matchExpression(pattern,expression) matches a JME expression against a pattern. The pattern is also written in JME syntax, but there are extra operators available to allow extra control what does or doesn’t match.
The parameters of a commutative operation in the pattern (i.e. addition, multiplication, or equality) can match in any order. The algorithm matches greedily, reading from left to right in both the pattern to match and the input expression.
Patternmatching should only be used for assigning marks when the form of the student’s answer is what’s being assessed, for example when the student is asked to factorise a quadratic or reduce a fraction to lowest terms. For answers which are the result of a calculation, you should use the normal numerical marking algorithms because patternmatching can be too restrictive (or, if you’re not careful, too accepting!) In such cases, you could use patternmatching to provide feedback about possible errors the student made, when their answer is marked wrong by the numerical algorithm.
Patternmatching syntax¶

?
Match anything.

??
Match anything or nothing.

expr;g
Capture
expr
in the group namedg
.

m_any(expr1,expr2,...)
Match any of the expressions
exprN
.

m_all(expr)
Capture all terms (in an addition, multiplication, or other commutative operation) matching
expr
.

m_pm(expr)
Capture
expr
or(expr)
, i.e. plus or minus the given expression.

m_not(expr)
Match anything except
expr
.

m_uses(name1,name2,...)
Match any expression which uses the given variable names.

m_commute(expr)
Match the terms in
expr
in any order, following the laws of commutativity. (This is only required if you are usingmatchExpression
with thedoCommute
flag set tofalse
, and you only want to use commutativity in certain places)

m_nothing
Match nothing. Useful as an empty term to act as the righthand side of an addition, where you want to capture all terms in the lefthand side.

m_number
Match a single number.

m_type(type)
Match a single token of the given type. For example,
m_type(vector)
matches a vector, whilem_type(op)
matches any operator. See Data types for a list of data types.
To help with learning the new syntax, there is an online tool to test expressions against patterns at http://www.staff.ncl.ac.uk/christian.perfect/patternmatching/matching.html
Examples¶
Get all \(x\) terms in a polynomial:
m_all(m_pm(m_all(??)*m_any(x,x^?)));xs+m_all(??);rest
Get the coefficient and degree of an \(x\) term:
m_pm(m_all(??);coefficient*m_any(x,x^?;degree))
Get both sides of an equation:
?;left=?;right
Check \(x\) terms are collected on one side of an equation:
m_uses(x);xside = m_not(m_uses(x));otherside
Check that a quadratic is factorised:
(m_pm(??*x);a+?;b)*(m_pm(??*x);c+?;d)
Capture multiple powers of \(x\) and \(y\):
m_all( m_any( ??x, ??y, ??x^??, ??y^??, m_any(x,x^??)*m_any(y,y^??)*?? ) );terms + m_all(??;rest)