5 Metafunctions and Relations

▼ |  | Redex: Practical Semantics Engineering

1 | Patterns
2 | Terms
3 | Languages
4 | Reduction Relations
5 | Metafunctions and Relations
6 | Testing
7 | GUI
8 | Typesetting
| Index

On this page:

define-metafunction
define-metafunction/ extension
in-domain?
define-relation
current-traced-metafunctions

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### 5`` Metafunctions and Relations

All of the exports in this section are provided both by redex/reduction-semantics (which includes all non-GUI portions of Redex) and also exported by redex (which includes all of Redex).

( define-metafunction language
contract
[(name pattern ...) term extras ...]
...)

contract

=

|  | | |  | id : pattern ... -> pattern
extras |  | = |  | (side-condition racket-expression)
|  | | |  | (side-condition/hidden racket-expression)
|  | | |  | (where tl-pat term)
|  | | |  | (where/hidden tl-pat term)
tl-pat |  | = |  | identifier
|  | | |  | (tl-pat-ele ...)
tl-pat-ele |  | = |  | tl-pat
|  | | |  | tl-pat ... ; a literal ellipsis

The define-metafunction form builds a function on sexpressions according to the pattern and right-hand-side expressions. The first argument indicates the language used to resolve non-terminals in the pattern expressions. Each of the rhs-expressions is implicitly wrapped in term.

The side-condition, hidden-side-condition, where, and where/hidden clauses behave as in the reduction-relation form.

Raises an exception recognized by exn:fail:redex? if no clauses match, if one of the clauses matches multiple ways (and that leads to different results for the different matches), or if the contract is violated.

Note that metafunctions are assumed to always return the same results for the same inputs, and their results are cached, unless caching-enabled? is set to #f. Accordingly, if a metafunction is called with the same inputs twice, then its body is only evaluated a single time.

As an example, these metafunctions finds the free variables in an expression in the lc-lang above:

( define-metafunction lc-lang
free-vars : e -> (x ...)
[(free-vars (e_1 e_2 ...))
(∪ (free-vars e_1) (free-vars e_2) ...)]
[(free-vars x) (x)]
[(free-vars ( lambda (x ...) e))
(- (free-vars e) (x ...))])

The first argument to define-metafunction is the grammar (defined above). Following that are three cases, one for each variation of expressions (e in lc-lang). The free variables of an application are the free variables of each of the subterms; the free variables of a variable is just the variable itself, and the free variables of a lambda expression are the free variables of the body, minus the bound parameters.

Here are the helper metafunctions used above.

( define-metafunction lc-lang
∪ : (x ...) ... -> (x ...)
[(∪ (x_1 ...) (x_2 ...) (x_3 ...) ...)
(∪ (x_1 ... x_2 ...) (x_3 ...) ...)]
[(∪ (x_1 ...))
(x_1 ...)]
[(∪) ()])
( define-metafunction lc-lang
- : (x ...) (x ...) -> (x ...)
[(- (x ...) ()) (x ...)]
[(- (x_1 ... x_2 x_3 ...) (x_2 x_4 ...))
(- (x_1 ... x_3 ...) (x_2 x_4 ...))
(side-condition ( not ( memq ( term x_2) ( term (x_3 ...)))))]
[(- (x_1 ...) (x_2 x_3 ...))
(- (x_1 ...) (x_3 ...))])

Note the side-condition in the second case of -. It ensures that there is a unique match for that case. Without it, ( term (- (x x) x)) would lead to an ambiguous match.

( define-metafunction/extension f language
contract
[(g pattern ...) term extras ...]
...)

Defines a metafunction g as an extension of an existing metafunction f. The metafunction g behaves as if f’s clauses were appended to its definition (with occurrences of f changed to g in the inherited clauses).

For example, define-metafunction/extension may be used to extend the free-vars function above to the forms introduced by the language lc-num-lang.

( define-metafunction/extension free-vars lc-num-lang
free-vars-num : e -> (x ...)
[(free-vars-num number)
()]
[(free-vars-num (+ e_1 e_2))
(∪ (free-vars-num e_1)
(free-vars-num e_2))])

( in-domain? (metafunction-name term ...))

Returns #t if the inputs specified to metafunction-name are legtimate inputs according to metafunction-name’s contract, and #f otherwise.

( define-relation language
[(name pattern ...) term ...] ...)

tl-pat |  | = |  | identifier
|  | | |  | (tl-pat-ele ...)
tl-pat-ele |  | = |  | tl-pat
|  | | |  | tl-pat ... ; a literal ellipsis

The define-relation form builds a relation on sexpressions according to the pattern and right-hand-side expressions. The first argument indicates the language used to resolve non-terminals in the pattern expressions. Each of the rhs-expressions is implicitly wrapped in term.

Relations are like metafunctions in that they are called with arguments and return results (unlike in, say, prolog, where a relation definition would be able to synthesize some of the arguments based on the values of others).

Unlike metafunctions, relations check all possible ways to match each case, looking for a true result and if none of the clauses match, then the result is #f. If there are multiple expressions on the right-hand side of a relation, then all of them must be satisfied in order for that clause of the relation to be satisfied.

Note that relations are assumed to always return the same results for the same inputs, and their results are cached, unless caching-enable? is set to #f. Accordingly, if a relation is called with the same inputs twice, then its right-hand sides are evaluated only once.

( current-traced-metafunctions) → ( or/c 'all ( listof symbol?))
( current-traced-metafunctions traced-metafunctions) → void?
traced-metafunctions : ( or/c 'all ( listof symbol?))

Controls which metafunctions are currently being traced. If it is 'all, all of them are. Otherwise, the elements of the list name the metafunctions to trace.

Defaults to '().

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