# Pattern-matching mathematical expressions¶

Page status: Draft

## Problem¶

In Numbas, we have two related tasks: give feedback based on the form of a student’s answer, and “simplify” expressions according to a configurable set of rules.

Both of these tasks involve pattern-matching in some form: to give feedback, different forms of algebraically equivalent expressions must be described in some general way; to simplify an expression, a rewriting rule must recognise expressions in a certain form and identify particular sub-expressions to produce a new form.

Pattern-matching on trees is not as simple as on strings.

If you’ve done some coding, you might have encountered regular expressions: these are a standard way of describing patterns in text strings, often used to perform text replacements. Regular expressions aren’t appropriate for mathematical expressions, because there’s a lot of structure that regular expressions can’t pick up. The most obvious of these is that regular expressions can’t deal with matching brackets. For example, in the simplification rule $$a \times (-b) \to -(a \times b)$$, $$a$$ and $$b$$ are arbitrary expressions which may contain any number of nested brackets. There’s also a wealth of semantic information about mathematical operations, such as commutativity and associativity, that a pattern-matching system should know about and take into account.

So, we need a pattern-matching system which operates on mathematical expressions with all of their structure, not strings. This isn’t a new idea: most computer algebra systems, such as Mathematica and Maple, are fundamentally pattern-matching and term-rewriting systems.

The problem is that how the algorithms in those systems work on a technical level isn’t well documented, and because Numbas runs entirely on the client, we need an algorithm written in JavaScript.

Thanks to a very fruitful discussion with a couple of attendees at EAMS 2018, I came up with a rough description of the tools that a good pattern-matching system should offer.

I’ve spent the time since then implementing the system, and in using it I came up with some more features that greatly widened the range of possible uses.

Things we want:

• Consider all terms of an associative operation together, e.g. terms in a sum, joined by $$+$$ or $$-$$.

• Match if a subset of the terms match the given pattern, leaving the other terms unmatched.

• Accept several different forms with the same rule, to avoid repetition, e.g. $$x+By$$ is a special case of $$Ax+By$$ with $$A=1$$.

• It would be nice to have built-in matchers for common non-atomic objects, e.g. rational polynomials, complex numbers in Cartesian or polar form, diagonal matrices.

• A fairly readable (and writable) syntax for defining patterns which match expressions.

It doesn’t need to make immediate sense, but shouldn’t be too complicated: regular expressions don’t make sense to someone who hasn’t seen them before, but there are very few things to learn.

• Capture a missing term with a default value.

• Identify equivalent terms that appear in more than one place, for collecting or cancelling.

Once you’ve got pattern-matching and rewriting, you can do lots of things:

• Check the form of an answer.

• Simplify expressions.

• Symbolic differentiation and integration.

• Obtain canonical forms for (some classes of) expressions.

• Other CAS techniques.

### Rewriting an expression¶

To define a rewriting rule, we need a pattern that the expression must match and a result expression, which might have parts of the matched expression substituted into it.

If the language for patterns is not expressive enough, there could be combinatorial explosion in the number of rules that must be defined. (But a more expressive pattern language could lead to combinatorial explosion in the algorithm’s running time).

Rewriting rules should ignore parts of the expression that remain unchanged, and keep them roughly where they were in the rewritten expression.

It must be possible to evaluate a captured term to an object that you can do arithmetic with, for example collecting together constant terms.

It’s often much easier to use several rules together, rather than writing one rule which does everything. For example, when simplifying the fraction $$\frac{18}{6}$$, one rule could first cancel the common factor to obtain $$\frac{3}{1}$$, and then another rule could remove the denominator $$1$$ to obtain $$3$$.

#### Examples¶

• Take negation out of a fraction:

$$\frac{-x}{y}$$ becomes $$- \frac{x}{y}$$.

• Collect constants:

$$1 + x + 3$$ becomes $$x + 4$$.

• Collect coefficients of arbitrary terms:

$$5 \times (x + \sin(z)) - 3 \times (x+\sin(z))$$ becomes $$2 \times (x+\sin(z))$$.

• Remove terms with a factor of zero:

$$\cos(t) + 0 \times e^{5t} + z$$ becomes $$\cos(t) + z$$.

• Simplify square roots of square numbers:

$$\sqrt{16}$$ becomes $$4$$, but $$\sqrt{3}$$ is unchanged.

• Apply trigonometric identities:

$$\cos(\pi/2)$$ becomes $$0$$, $$\sin(3\pi/2)$$ becomes $$-1$$, but $$\sin(0.34 \pi)$$ is unchanged.

• Cancel common factors on the top and bottom of a fraction:

$$\displaystyle \frac{4 \, a^2 \, b \, c}{6 \, a \, b}$$ becomes $$\displaystyle \frac{2 \, a \, c}{3}$$.

• Extract a scalar factor from the elements of a matrix:

$$\begin{pmatrix} 2\lambda & 0 \\ 0 & - \lambda \, x \end{pmatrix}$$ becomes $$\lambda \begin{pmatrix} 2 & 0 \\ 0 & -x \end{pmatrix}$$.

### Determining the form of an expression¶

In a ‘mathematical expression’ question, the student’s answer $$S$$ is an algebraic expression which must match the teacher’s expected answer $$T$$.

We can numerically establish that the student has given an equivalent expression by substituting values for $$x$$ into both $$S$$ and $$T$$, but this tells us nothing about the form of $$S$$.

Often, valid forms of answer to a particular question differ substantially from each other. The order of terms in a sequence of commuting operations usually doesn’t matter, and some terms may be optional or not present in some randomisations of the question.

#### Examples¶

• The student must expand $$(x+\alpha)(x+\beta)$$.

The expanded expression will be of the form $$x^2+Ax+B$$.

Necessary conditions for ‘expanded’ include ‘contains no brackets’, and ‘each term is an integer multiplied by a power of $$x$$’.

The $$x$$ and constant terms might be omitted.

If the coefficient $$A = 1$$, it can be omitted.

The opposite task, factorise $$x^2+Ax+B$$, will produce an expression of the form $$(x+\alpha)(x+\beta)$$.

A condition for ‘factorised’ is ‘the expression is a product of irreducible polynomials’.

• Write a complex number in argument-modulus form, $$r\,e^{i\theta}$$ - $$r$$ and $$\theta$$ could be literal real numbers, or expressions producing real numbers.

If either of $$r$$ or $$\theta$$ are $$1$$, they can be omitted.

If $$\theta = 0$$, then the whole exponent could be written as $$e^0$$, $$1$$, or omitted entirely.

Some complex numbers in argument-modulus form:

• $$5e^{-2i}$$

• $$5e^{3i}$$

• $$e^i$$

• $$(1+\sqrt{2})e^{\frac{\pi}{2}i}$$

• $$1.32445e^0$$

• $$1$$

## Previous work¶

### WeBWorK¶

WeBWorK uses ‘bizarro arithmetic’ to force expressions which would be equivalent in standard arithmetic to be non-equivalent. It then uses the trick of evaluating at randomly chosen points to establish equivalence. Still unable to give reliable feedback on the form of the student’s answer. Quite a lot of work to set it all up (add flags to context, etc.)

limitedFactor context

bizarro math for sine and cosine - Davide suggests directly inspecting the Formula object to test if it’s of the form $$\sin(\cdot)$$.

Adaptive parameters try to allow for free variables which change the value of an expression linearly, i.e. $$Af(x) + B$$ instead of $$f(x)$$. Another randomised algorithm is used to establish how the parameters affect the expression, as a matrix - pick some random values for the parameters, and solve the resulting system of equations.

### Prolog¶

(Because Chris Sangwin told me to look at it).

Prolog uses a variant of the Martelli-Montanari unification algorithm to identify values of free variables on either side of an equation so that they are equivalent.

It doesn’t allow for missing values, or alternate forms in one expression - you’d have to give an equation for each form.

### STACK¶

STACK has a few answer tests to do with the form of the student’s answer: LowestTerms, Expanded, FacForm, SingleFrac, PartFrac, CompletedSquare.

For anything else, you can apply simplification rules to expressions before comparing - the two expressions should end up exactly equal after simplifying.

### Maxima¶

Maxima deals with everything as S-expressions, and seems to require quite a lot of code to add new rules.

### expreduce, Mathics, Mathematica¶

expreduce is a project in Go. Inspired by Mathics, but some syntax differences.

A video by Brian Beckman about how term rewriting in the style of Mathematica works. Jacquard is a JavaScript clone of Mathematica’s syntax.

Mathics pattern-matching seems to have many of the same operators I’ve come up with. Turns out it’s basically a clone of Mathematica.

Mathematica‘s functionality is similar to what I came up with.

### Maple¶

Maple’s pattern matching commands don’t look as sophisticated as Mathematica, but there are some shorthands for common patterns: algebraic, linear, multilinear.

### Matchpy¶

Matchpy is a library for pattern-matching symbolic expressions in Python.

It seems to be inspired by Mathematica, and isn’t very sophisticated at the moment.

It has a “many-to-one “matcher which tries to match an expression against several patterns at once. Where patterns are similar, it only needs to check once. Could this be modelled with a nondeterministic finite state automaton?

Rewriting rules are implemented as patterns, with a Python function whose parameters are the captured names, returning a transformed expression.

### Rubi¶

Rubi symbolically integrates.

It’s a collection of rewriting rules, first implemented in Mathematica and later ported to other computer algebra systems.

## Solution¶

To define patterns, I added several new symbols to the JME language, for quantifiers, conjunctions and atoms.

I originally had lots of functions of the form m_X, which led to quite long and hard-to-read patterns. Operators are a lot easier to read, especially the postfix quantifiers which look more like regex quantifiers, e.g. x ? rather than m_maybe(x).

The pattern-matching algorithm takes a pattern written with these operators and an input expression, and decides if there’s a match, producing a collection of captured parts analogous to regular expression capturing groups.

### Similarities with regular expressions.¶

The matching algorithm is backtracking, like many implementations of regular expression matchers. When there is a choice to make, due to a quantifier or the order of terms, the matcher can backtrack and make a different choice if one path turns out not to lead to a match.

Quantifiers allow arbitrarily many similar terms to be captured without repetition in the pattern.

The & operator for specifying two patterns that an expression must match is analogous to positive lookahead in regular expressions.

Unlike most regular expression matchers, patterns can specify arbitrarily complicated conditions on parts of the expression in the middle of the matching process. For example, a regular expression can’t test that a number is prime in one pass.

### Syntax trees¶

A string representing a mathematical expression or pattern is parsed into a syntax tree.

The pattern-matcher operates on these trees.

class Tree

A tree consists of a token, which has a type, and an ordered list of arguments, which are themselves trees.

The types include, but are not limited to:

• Function application

• Operator (binary, unary prefix or unary suffix)

• Variable name

• Number

So $$\sin(x)+1$$ would be:

operator '+'
function 'sin'
variable x
number 1


### Options¶

Commutative

When matching terms joined by a commutative operation such as $$\times$$ or $$+$$, match terms in any order.

Associative

When matching terms joined by an associative operation, collect as many terms as possible to match at once, instead of just the two subtrees of the first application of the operation. e.g., $$(a+b)+c$$ is matched as a list of three terms $$a$$, $$b$$, $$c$$, not two terms $$a+b$$ and $$c$$.

Allow other terms

Match a sequence of terms where the pattern is satisfied by a subset of the terms. e.g., $$1+2+x$$ matches n +n - the extra $$x$$ is ignored. In a non-commutative match, the pattern must match a contiguous subsequence of the terms.

Strict inverse

If turned off, x-y will be considered as x+(-y), so will match patterns like ?+?.

Similar for x/y being interpreted as x*(/y). If turned on, plus means plus!

Gather as a sequence

If turned off, then multiple terms matched under the same name will be stored in a list.

If turned on, then they will be captured as a sequence of terms joined by the same operator used to find them, e.g. addition or multiplication.

### Matching a pattern¶

matchTree(ruleTree, exprTree)

To decide whether the given exprTree matches the given ruleTree, and return any matched names if so.

This algorithm operates recursively. Its behaviour depends on the type of the token at the top of ruleTree:

These produce either false if there is no match, or a match object, mapping captured names to the corresponding parts of exprTree.

Most of these cases are quite easy. The hardest task is to match a sequence of terms - that’s where quantifiers come in.

matchName(ruleTree, exprTree)

When the ruleTree is a single name token, it is either a special name representing some built-in pattern, or a generic name.

If the name is not special, it matches exprTree if exprTree is also exactly that single name token.

The special names are:

• ? - matches any expression.

• $n - matches a literal number. Annotations on this can specify extra conditions, or match other kinds of numbers, such as fractions or complex numbers. • $v - matches any variable name.

• $z - never matches. Used as a dummy term when matching sequences of terms, all of which are matched by a single pattern. matchFunction(ruleTree, exprTree) Match the application of a function. ruleTree is a function application token, with an ordered list of arguments. If it’s a special function, run the logic for that. Otherwise, match as an ordinary function: • If the expression is not a function application, return false. • If the name of the function in the expression is not the same as the one in the rule and the rule function’s name is not ?, return false. • Consider the arguments of both function applications as a sequence, and run matchTermSequence(). Collect names matched in the arguments as follows: if a name is only matched by one argument, keep the match as returned by matchTermSequence. If it’s matched by more than one argument, the name is matched to a list, with an entry for each argument the name was captured in. The special matching functions either change options for the sub-pattern they enclose, or specify conditions: • m_uses(names) - Matches if exprTree uses all of the given names as free variables. • m_exactly - Turns off the “Allow other terms” option. • m_commutative - Turns on the “Commutative” option. • m_noncommutative - Turns off the “Commutative” option. • m_associative - Turns on the “Associative” option. • m_nonassociative - Turns off the “Associative” option. • m_strictinverse - Turns on the “Strict” option. • m_gather - Turns on the “Gather as a sequence” option. • m_nogather - Turns off the “Gather as a sequence” option. • m_type(type) - Matches if the token at the top of exprTree is of the given type. • m_func(name,arguments) - Matches if exprTree is the application of the function with the given name, and its arguments match the given list. • m_op - Matches if exprTree is the application of the operator with the given name, and the operands read from left to right match the given list. • m_anywhere(subpattern) - Matches if the subpattern matches anywhere within exprTree - perform a breadth-first search of exprTree, returning the first match. matchOp(ruleTree, exprTree) Match the application of an operator. ruleTree is an operator token, with a list of operands. If it’s a special operator, run the logic for that. Otherwise, match an ordinary operator: The operator being matched is the operator at the top of ruleTree. The match is commutative if the commutative option is turned on and the operator is commutative. The match is associative if the associative option is turned on and the operator is associative. Run getTerms() to identify terms in ruleTree and exprTree. If the match is not associative, and the expression is not an application of the operator being matched, and it’s a unary operation, then there is no match. Run matchTermSequence() on the terms. Unmatched terms are allowed if the “Allow other terms” option is turned on and the match is associative. Collate the named groups: for names which are matched more than once, combine them. If the “Gather as a sequence” option is turned off, the name matches to a list with an entry for each time the name was matched. If it’s turned on, the occurrences are joined together by applications of the operator. When gathering multiplicative terms as a sequence, the invented unary reciprocal operator must be removed: replace each instance of x*(/y) with x/y. Capture the operator token under the name __op__, to be used by a rewriting rule if there are unmatched terms in the sequence. The special matching operators specify quantifiers, allow for plus/minus or times/divide matches, or express combinations of patterns: Quantifiers: • subpattern? - match subpattern if possible, otherwise ignore it. • subpattern* - match any number of terms matching subpattern. • subpattern+ - match one or more terms matching subpattern. Combining patterns: • a | b - “either a or b”. If exprTree matches pattern a, return that, otherwise try to match b. • ! subpattern - “not subpattern”. exprTree only matches if it does not match subpattern. • a & b - “both a and b”. Test both a and b, and combine their matched names. Conditions: • subpattern where condition. exprTree must match subpattern, and then after substituting matched names into condition, it must evaluate to true. Inverses: • +- subpattern. “Plus or minus subpattern”. exprTree must match either subpattern, or -(subpattern). • */ subpattern. “subpattern or its reciprocal”. exprTree must match either subpattern, or /(subpattern). The unary reciprocal operator is added when collecting the terms in a sequence, replacing x/y with x*(/y). Other special operators: • subpattern : v. “Default value for missing term”. When matching a sequence, if subpattern is not matched, then this term is matched as the default value v instead. • macros @ subpattern. macros is a dictionary mapping names to patterns. The macros are substituted into subpattern before running matchTree() to find a match. matchList(ruleTree, exprTree) ruleTree has a list token at the top, and its arguments are a sequence of patterns. If the token at the top of exprTree is not a list, return false. Consider the elements of both lists as a sequence of terms. Run matchTermSequence() on the terms. Each matched name is captured as a list with an entry for each time the name was matched. matchToken(ruleTree, exprTree) There is a match if the tokens at the top of ruleTree and exprTree are equal. #### Matching a sequence of terms¶ Function arguments, list elements, and the operands of associative operations are considered as sequences of terms. Both the pattern and the expression being matched produce a sequence of terms. The aim is to match up terms in the pattern with terms in the expression. Quantifiers on each term in the pattern specify how many terms in the expression can match against it. To identify a sequence, we might need to apply the law of associativity for binary operations. When matching a sequence pattern we might need to apply the law of commutativity, to match terms which appear in a different order to that used in the pattern. It’s convenient when matching sums or products of terms to treat x-y as x+(-y) and x/y as x*(/y). There’s no conventional symbol for a unary “reciprocal” operator analogous to the unary “negation” operator, but it’s useful here. class Term(tree) A Term objects represents a single term in a sequence. The same Term class is used for terms in both the pattern and the expression. Terms in the expression only fill in the tree property; the rest of the defined properties are only for terms in the pattern. property tree: Tree The syntax tree corresponding to this term, with the outermost quantifiers, default value operators, or capturing operators removed. property names Names under which this term should be captured. property inside_equalnames Names captured inside a quantifier that have an equality condition - each term matching this term will be considered individually for equality. property outside_equalnames Names captured outside a quantifier that have an equality condition - all terms matching this term will be grouped together, and then any later matches to the same name will be considered for equality with these as a single expression. property quantifier How many terms in the expression can match against this pattern term. Possible values: • 0 - this term must not match. • 1 - match once • ? - match once or not at all. • * - match any number of times. • + - match at least once. property defaultValue A value to capture if this term is not matched. To compute these values, any special operators on the outside of the input tree are peeled off, one by one: The initial value of quantifier is 1, or 0 if the term is $z.

Quantifiers are pulled through unary operations, so -(x?) is equivalent to (-x)?.

;

Add a name to names.

;=

Add a name to names, and also add it to outside_equalnames if a quantifier has not been encountered yet, or inside_equalnames otherwise.

?, * or +

Change quantifier.

There are precedence relations between the current value of quantifier and the one being unpeeled:

• 0 takes precedence over all others.

• 1 has the lowest precedence.

• ? followed by * or +, or the other way round, produce *.

• Otherwise, the new quantifier takes precedence.

: - set defaultValue.

If quantifier is 1, change it to ?. If it’s +, change it to *. (Implicitly, this term is optional)

Once these operators have been peeled off, the remaining tree is saved as the tree property.

Finally, any more identified capturing operators ;= inside the tree are found and saved. Those under a quantifier are ignored.

getTerms(tree, op, commutative, associative, strictInverse)

Given a tree representing a series of terms t1 op t_2 op t_3 op ... op t_n, return the terms as a list of Term objects.

• If the top of the tree is a unary minus operation, move it inside the tree so that it applies to the leftmost factor. For example, rewrite -((x*y)*z) to ((-x)*y)*z.

• If the “Strict inverse” option is not turned on, and op is + or *, then replace x-y with x+(-y) and x/y with x*(/y).

• If op is a binary relation, then we want to make sure that the operation and its converse are handled together. For example, > is the converse of <. If op is <, then replace all instances in the tree of a > b with b < a.

• If the token at the top of the tree is the operator op, then args is the list of arguments. Otherwise, args is the list whose only element is tree.

• For each argument arg in args:

• Make a Term object term containing arg.

• Remove any capturing operators ; or ;= from the top of arg.

• If op is * and the token at the top of arg is a unary minus, then remove that (the Term object will remember that it’s there).

• If using associativity and the token at the top of arg is the operator op, then split it into more terms:

• Run getTerms() on arg to obtain a list of terms.

• Add the captured names from item to each of these terms.

• If the quantifier for item is not 1, then combine the quantifiers on each of these terms, using the same logic as used when constructing Term objects. For example, (x+ * y)? is equivalent to x* * y?.

• Add each of these terms to the list of terms to output.

Otherwise, add term to the list of terms to output.

• Return the list of identified terms.

matchTermSequence(ruleTerms, exprTerms, options)

Given a list of rule terms and a list of expression terms, try to come up with an assignment of expression terms to rule terms.

A rule term might match more than one expression term, if quantifiers allow.

An expression term might not match any rule terms, if the “Allow other terms” option is turned on. In order to allow rewriting rules to keep terms in roughly the same order, we track whether unmatched expression terms are towards the start or the end of the expression.

Keep track of what names have been matched for each term in the expression.

An expression term exprTerm matches a rule term ruleTerm if matchTree(exprTerm.tree, ruleTerm.tree)() returns true.

An assignment is valid if, for each identified name captured in an expression term, it’s equal to all matches of the name in previous terms.

There might be more than one valid matching of expression terms to rule terms: an expression term could match several rule terms, and a rule term might have a quantifier that allows it to match different numbers of expression terms. If “Allow other terms” is turned on, then for each expression term we could decide not to try to match it at all.

The strategy is to move the input pointer along the list of expression terms, trying to match them greedily with the first rule term they match against. If either pointer reaches the end of the corresponding list and there are any unmatched terms, then backtrack, either looking to match an expression term against a later rule, or if “Allow other terms” is turned on, don’t match it at all.

When the “Commutative” option is turned on, there are lots more possible matches, increasing the maximum running time of the algorithm.

The process of finding a match works as follows.

Define:

capture

A list storing which pattern term each expression term is matched against. Values are either an index in the list ruleTerms or the constants UNCAPTURED, START or END, denoting a term that did not match a rule term, not captured, at the start of the sequence or at the end. Initially, every value in this list is UNCAPTURED.

start

The index of the first term in the expression to consider matching, initially 0, meaning the first term in the list. Terms before this are not included in the match.

pc

‘Pattern pointer’ - a pointer into the list of rule terms. Initially 0, meaning the first term.

ic

‘Input pointer’ - a pointer into the list of expression terms. Initially 0, meaning the first term.

Consumed

A rule term is consumed if its quantifiers allow no more expression terms to be matched against it.

Enough

A rule term has had enough if the number of expression terms matched against it is at least the minimum required by its quantifiers.

To find a match, repeat the following loop until done:

Algorithm (Main loop)
• Move to the next unconsumed rule term: as long as the rule term pointed to by pc is consumed, increment pc.

• If the input pointer ic has reached the end of the expression terms:

• Move to the next rule term that hasn’t had enough: while the rule term at pc has had enough, increment pc.

• If pc has reached the end of the list of rule terms:

• If there is any rule term that has not had enough, Backtrack.

• Otherwise, this is a match.

• Otherwise, there are still some unmatched rule terms, so Backtrack.

• Otherwise, if the pattern pointer pc has reached the end of the rule terms, there are unconsumed expression terms:

• If “Allow other terms” isturned on:

• If “Commutative” is turned on, then mark the expression term pointed to by ic as END, and Advance input.

• Otherwise, capture all the remaining expression terms as END, and move the input pointer ic to the end.

• Otherwise, Backtrack.

• Otherwise, check if the expression term pointed to by ic matches the rule term pointed to by pc.

If it does, and all captured name assignments are valid, capture ic as matching pc and Advance input.

• Otherwise, this expression term doesn’t match the current rule term. If “Commutative” is turned on, or if the rule terms pointed to by pc has had enough, increment pc.

• If none of those apply, Backtrack.

Algorithm (Backtrack)

Find the last place we made a choice:

• If “Allow other terms” is turned on, ic = start, the number of captured terms is equal to start and less than the number of input terms, then try ignoring the term pointed to by ic:

Capture the expression term pointed to by ic as START, and Increment start.

• Otherwise:

• Move ic back at least one place, to the last term before start or the last term not captured as END or UNCAPTURED, whichever is last.

• If ic < start:

• If “Allow other terms” is turned on and start is not at the end:

• Capture all terms before start as START.

• Otherwise, there is no match: no expression terms have matched any rule terms.

• Set pc to the position after the one which the expression term pointed to by ic is captured as (or just the end, if the term at ic is captured as END).

• Mark all expression terms after ic as UNCAPTURED.

Algorithm (Increment start)
• Set ic = start, so we ignore expression terms before that.

• Set pc = 0, since we’re effectively starting the matching process again with a smaller list of expression terms.

• Increment ic.

• If “Commutative” is turned on, set pc = 0, so that each expression term can eventually be tried against every rule term.

If a match is found, then we need to match up captured names from each term to produce a final match result.

For each expression term exprTerm matched against a rule term ruleTerm, record that exprTerm was matched against each of the names captured by ruleTerm, and merge this with any names captured inside ruleTerm.

For each rule term with a Term.defaultValue that did not match any expression terms, match the default value against all the rule term’s capturing names.

Record expression terms captured as START or END in separate lists, as well as the operator. These will be used in rewrite().

### Simplification rules¶

A simplification rule is a term rewriting rule $$l \to r$$. For example, $$x \times (y \times z) \to (x \times y) \times z$$ changes the order in which a product of three terms is evaluated. In this instance, $$x$$, $$y$$ and $$z$$ are arbitrary sub-expressions.

If this rule was applied repeatedly to an arbitrarily bracketed product of several terms, the final expression would end up looking like:

$(((\ldots (t_1 \times t_2) \times t_3) \ldots ) \times t_n)$
simplify(rules, exprTree)

Apply this loop:

• First, simplify all arguments of $$exprTree$$.

• Find the first rule r in rules that matches exprTree.

If there is none, exit the loop, returning the latest version of exprTree.

• Use r to rewrite exprTree.

rewrite(ruleTree, resultTree, exprTree)

Match exprTree against ruleTree.

If there’s no match, return exprTree unchanged.

We now have a set of captured names, and a corresponding sub-expression for each of them constructed from parts of exprTree or default values defined in the rule.

For names defined in ruleTree that did not capture anything, associate them with a ‘nothing’ value.

For each name captured in the match, substitute the corresponding values into resultTree.

Apply post-replacement rules:

• Replace eval(expr) with the result of evaluating expr.

• Replace binary operations where one argument is ‘nothing’ with just the other argument, i.e. replace nothing (op) x and x (op) nothing with x.

If the match has any expression terms that were ignored in matchTermSequence(), add these: using the recorded operator op, ignored terms rest_start and rest_end, and the rewritten tree result, produce rest_start (op) result (op) rest_end.

This is a naive algorithm - with the wrong set of rules, it can get stuck in an infinite loop.

## Problems¶

“Identified names”, when interacting with commutative match, needs some kind of backtracking over trees. For example, in order for ?*?;=y + ?*?;=y to match 3*x + x*5, each of the additive terms needs to know about the others while matching the product operation. It’s not enough just to tell the second term that the name $$y$$ has been matched to $$x$$ in the first term: it might be that the first term needs to match differently.

So the identified names should maybe be applied at the very end, to the whole expression, but we then need a way of asking for “the next” way of matching each set of terms.