On this page:
4.1 Patterns
redex-match
redex-match?
match?
match-bindings
bind
caching-enabled?
set-cache-size!
check-redundancy
4.2 Terms
term
hole
in-hole
mf-apply
term-let
redex-let
redex-let*
define-term
term-match
term-match/  single
plug
variable-not-in
variables-not-in
exn:  fail:  redex?
4.3 Languages
define-language
4.3.1 Binding Forms
4.3.2 Multiple Variables in a Single Scope
4.3.3 Ellipses in Binding Forms
4.3.4 Compound Forms with Binders
4.3.5 Binding Repetitions
:  :  =
shadow
nothing
define-extended-language
define-union-language
make-immutable-binding-hash
make-binding-hash
language-nts
compiled-lang?
default-language
alpha-equivalent?
substitute
4.4 Reduction Relations
reduction-relation
extend-reduction-relation
union-reduction-relations
reduction-relation->rule-names
compatible-closure
context-closure
reduction-relation?
apply-reduction-relation
apply-reduction-relation/  tag-with-names
apply-reduction-relation*
current-cache-all?
-->
fresh
with
4.5 Other Relations
define-metafunction
define-metafunction/  extension
in-domain?
define-judgment-form
define-extended-judgment-form
judgment-holds
build-derivations
derivation
I
O
define-relation
judgment-form?
IO-judgment-form?
current-traced-metafunctions
4.6 Testing
test-equal
test-->>
test-->
test-->>∃
test-->>E
test-judgment-holds
test-predicate
test-results
default-equiv
make-coverage
coverage?
relation-coverage
covered-cases
generate-term
redex-enum
redex-index
redex-check
depth-dependent-order?
redex-generator
counterexample
exn:  fail:  redex:  test
check-reduction-relation
check-metafunction
default-attempt-size
default-check-attempts
redex-pseudo-random-generator
exn:  fail:  redex:  generation-failure?
4.7 GUI
traces
traces/  ps
stepper
stepper/  seed
show-derivations
derivation/  ps
term-node-children
term-node-parents
term-node-labels
term-node-set-color!
term-node-color
term-node-set-red!
term-node-expr
term-node-set-position!
term-node-x
term-node-y
term-node-width
term-node-height
term-node?
reduction-steps-cutoff
initial-font-size
initial-char-width
dark-pen-color
dark-brush-color
light-pen-color
light-brush-color
dark-text-color
light-text-color
pretty-print-parameters
default-pretty-printer
4.8 Typesetting
4.8.1 Picts, PDF, & Post  Script
render-term
term->pict
render-term/  pretty-write
term->pict/  pretty-write
render-language
language->pict
render-reduction-relation
reduction-relation->pict
render-metafunction
render-metafunctions
metafunction->pict
metafunctions->pict
render-relation
render-judgment-form
relation->pict
judgment-form->pict
4.8.2 Customization
render-language-nts
non-terminal-gap-space
extend-language-show-union
extend-language-show-extended-order
render-reduction-relation-rules
rule-pict-style
reduction-rule-style/  c
rule-pict-info?
rule-pict-info-arrow
rule-pict-info-lhs
rule-pict-info-rhs
rule-pict-info-label
rule-pict-info-computed-label
rule-pict-info->side-condition-pict
arrow-space
label-space
metafunction-pict-style
metafunction-up/  down-indent
delimit-ellipsis-arguments?
white-square-bracket
homemade-white-square-bracket
default-white-square-bracket
linebreaks
sc-linebreaks
metafunction-cases
judgment-form-cases
judgment-form-show-rule-names
label-style
grammar-style
paren-style
literal-style
metafunction-style
non-terminal-style
non-terminal-subscript-style
non-terminal-superscript-style
default-style
label-font-size
metafunction-font-size
default-font-size
reduction-relation-rule-separation
reduction-relation-rule-extra-separation
reduction-relation-rule-line-separation
curly-quotes-for-strings
current-text
arrow->pict
set-arrow-pict!
white-bracket-sizing
horizontal-bar-spacing
metafunction-gap-space
metafunction-rule-gap-space
metafunction-line-gap-space
metafunction-fill-acceptable-width
metafunction-combine-contract-and-rules
relation-clauses-combine
metafunction-arrow-pict
where-make-prefix-pict
where-combine
current-render-pict-adjust
4.8.3 Removing the Pink Background
with-unquote-rewriter
with-atomic-rewriter
with-atomic-rewriters
with-compound-rewriter
with-compound-rewriters
4.8.4 LWs
lw
build-lw
to-lw
to-lw/  stx
render-lw
lw->pict
just-before
just-after
fill-between
4.8.5 Macros and Typesetting

4 The Redex Reference

 (require redex) package: redex-gui-lib

The redex library provides all of the names documented here.

Alternatively, use the redex/reduction-semantics and redex/pict libraries, which provide only non-GUI functionality (i.e., everything except redex/gui), making them suitable for programs that should not depend on racket/gui/base.

4.1 Patterns

 (require redex/reduction-semantics) package: redex-lib

This section covers Redex’s pattern language, which is used in many of Redex’s forms. Patterns are matched against terms, which are represented as S-expressions.

Pattern matching uses a cache—including caching the results of side-conditions—so after a pattern has matched a given term, Redex assumes that the pattern will always match the term.

In the following grammar, literal identifiers (such as any) are matched symbolically, as opposed to using the identifier’s lexical binding:

  pattern = any
  | _
  | number
  | natural
  | integer
  | real
  | string
  | boolean
  | variable
  | (variable-except id ...)
  | (variable-prefix id)
  | variable-not-otherwise-mentioned
  | hole
  | symbol
  | (name id pattern)
  | (in-hole pattern pattern)
  | (hide-hole pattern)
  | (side-condition pattern guard-expr)
  | (cross id)
  | (pattern-sequence ...)
  | other-literal
     
  pattern-sequence = pattern
  | ... ; literal ellipsis
  | ..._id

Changed in version 1.8 of package redex-lib: Non-terminals are syntactically classified as either always producing exactly one hole or may produce some other number of holes, and the first argument to in-hole is allowed to accept only patterns that produce exactly one hole.

syntax

(redex-match lang pattern term-expr)

(redex-match lang pattern)
If redex-match is given a term-expr, it matches the pattern (in the language) against the result of term-expr. The result is #f or a list of match structures describing the matches (see match? and match-bindings).

If redex-match has only a lang and pattern, the result is a procedure for efficiently testing whether terms match the pattern with respect to the language lang. The procedure accepts a single term and returns #f or a list of match structures describing the matches.

Examples:
> (define-language nums
    (AE number
        (+ AE AE)))
> (redex-match nums
               (+ AE_1 AE_2)
               (term (+ (+ 1 2) 3)))

(list (match (list (bind 'AE_1 '(+ 1 2)) (bind 'AE_2 3))))

> (redex-match nums
               (+ AE_1 (+ AE_2 AE_3))
               (term (+ (+ 1 2) 3)))

#f

> (redex-match nums
               (+ AE_1 AE_1)
               (term (+ (+ 1 2) 3)))

#f

syntax

(redex-match? lang pattern any)

(redex-match? lang pattern)
Like redex-match, but returns only a boolean indicating whether the match was successful.

Examples:
> (define-language nums
    (AE number
        (+ AE AE)))
> (redex-match? nums
                (+ AE_1 AE_2)
                (term (+ (+ 1 2) 3)))

#t

> (redex-match? nums
                (+ AE_1 AE_1)
                (term (+ (+ 1 2) 3)))

#f

procedure

(match? val)  boolean?

  val : any/c
Determines whether a value is a match structure.

procedure

(match-bindings m)  (listof bind?)

  m : match?
Returns a list of bind structs that binds the pattern variables in this match.

struct

(struct bind (name exp)
    #:extra-constructor-name make-bind)
  name : symbol?
  exp : any/c
Instances of this struct are returned by redex-match. Each bind associates a name with an s-expression from the language, or a list of such s-expressions if the corresponding name clause is followed by an ellipsis. Nested ellipses produce nested lists.

parameter

(caching-enabled?)  boolean?

(caching-enabled? on?)  void?
  on? : boolean?
When this parameter is #t (the default), Redex caches the results of pattern matching, metafunction, and judgment-form evaluation. There is a separate cache for each pattern, metafunction, and judgment-form; when one fills (see set-cache-size!), Redex evicts all of the entries in that cache.

Caching should be disabled when matching a pattern that depends on values other than the in-scope pattern variables or evaluating a metafunction or judgment-form that reads or writes mutable external state.

Changed in version 1.6 of package redex-lib: Extended caching to cover judgment forms.

procedure

(set-cache-size! size)  void?

  size : positive-integer?
Changes the size of the per-pattern, per-metafunction and per-judgment-form caches.

The default size is 63.

parameter

(check-redundancy)  boolean?

(check-redundancy check?)  void?
  check? : boolean?
Ambiguous patterns can slow down Redex’s pattern matching implementation significantly. To help debug such performance issues, set the check-redundancy parameter to #t. A true value causes Redex to, at runtime, report any redundant matches that it encounters.

Changed in version 1.9 of package redex-lib: Corrected spelling error, from check-redudancy to check-redundancy

4.2 Terms

Object language expressions in Redex are written using term. It is similar to Racket’s quote (in many cases it is identical) in that it constructs lists as the visible representation of terms.

The grammar of terms is (note that an ellipsis stands for repetition unless otherwise indicated):

  term = identifier
  | (term-sequence ...)
  | ,expr
  | (in-hole term term)
  | hole
  | (mf-apply identifier term ...)
  | datum
     
  term-sequence = term
  | ,@expr
  | ... ; literal ellipsis

syntax

(term term)

(term term #:lang lang-id)
Used for construction of a term.

The term form behaves similarly to quasiquote, except for a few special forms that are recognized (listed below) and that names bound by term-let are implicitly substituted with the values that those names were bound to, expanding ellipses as in-place sublists (in the same manner as syntax-case patterns).

The optional #:lang keyword must supply an identifier bound by define-language, and adds a check that any symbol containing an underscore in term could have been bound by a pattern in the language referenced by lang-id. In practice, this means that the underscore must be preceded by a non-terminal in that language or a built-in pattern such as number. This form of term is used internally by default, so this check is applied to terms that are constructed by Redex forms such as reduction-relation and define-metafunction.

For example,

(term-let ([body '(+ x 1)]
           [(expr ...) '(+ - (values * /))]
           [((id ...) ...) '((a) (b) (c d))])
  (term (let-values ([(id ...) expr] ...) body)))

evaluates to

'(let-values ([(a) +]
              [(b) -]
              [(c d) (values * /)])
   (+ x 1))

It is an error for a term variable to appear in an expression with an ellipsis-depth different from the depth with which it was bound by term-let. It is also an error for two term-let-bound identifiers bound to lists of different lengths to appear together inside an ellipsis.

Symbols in a term whose names end in guillemets (French quotes) around a number (for example asdf«5000») will be modified to contain a smiley face character (for example asdf«5000☺»). This is to prevent collisions with names generated by the freshening process that binding forms use.

syntax

hole

Recognized specially within terms. A hole form is an error elsewhere.

syntax

in-hole

Recognized specially within terms. An in-hole form is an error elsewhere.

syntax

mf-apply

Recognized specially within terms. A mf-apply form is an error elsewhere.

syntax

(term-let ([tl-pat expr] ...) body)

 
tl-pat = identifier
  | (tl-pat-ele ...)
     
tl-pat-ele = tl-pat
  | tl-pat ... ; a literal ellipsis
Matches each given id pattern to the value yielded by evaluating the corresponding expression and binds each variable in the id pattern to the appropriate value (described below). These bindings are then accessible to the term syntactic form.

Note that each ellipsis should be the literal symbol consisting of three dots (and the ... elsewhere indicates repetition as usual). If tl-pat is an identifier, it matches any value and binds it to the identifier, for use inside term. If it is a list, it matches only if the value being matched is a list value and only if every subpattern recursively matches the corresponding list element. There may be a single ellipsis in any list pattern; if one is present, the pattern before the ellipses may match multiple adjacent elements in the list value (possibly none).

This form is a lower-level form in Redex, and not really designed to be used directly. For let-like forms that use Redex’s full pattern matching facilities, see redex-let, redex-let*, term-match, term-match/single.

syntax

(redex-let language ([pattern expression] ...) body ...+)

Like term-let but the left-hand sides are Redex patterns, interpreted according to the specified language. It is a syntax error for two left-hand sides to bind the same pattern variable.

This form raises an exception recognized by exn:fail:redex? if any right-hand side does not match its left-hand side in exactly one way.

In some contexts, it may be more efficient to use term-match/single (lifted out of the context).

syntax

(redex-let* language ([pattern expression] ...) body ...+)

The let* analog of redex-let.

syntax

(define-term identifier term)

Defines identifier for use in term templates.

syntax

(term-match language [pattern expression] ...)

Produces a procedure that accepts term (or quoted) expressions and checks them against each pattern. The function returns a list of the values of the expression where the pattern matches. If one of the patterns matches multiple times, the expression is evaluated multiple times, once with the bindings in the pattern for each match.

When evaluating a term-match expression, the patterns are compiled in an effort to speed up matching. Using the procedural result multiple times to avoid compiling the patterns multiple times.

syntax

(term-match/single language [pattern expression] ...)

Produces a procedure that accepts term (or quoted) expressions and checks them against each pattern. The function returns the expression behind the first successful match. If that pattern produces multiple matches, an error is signaled. If no patterns match, an error is signaled.

The term-match/single form raises an exception recognized by exn:fail:redex? if no clauses match or if one of the clauses matches multiple ways.

When evaluating a term-match/single expression, the patterns are compiled in an effort to speed up matching. Using the procedural result multiple times to avoid compiling the patterns multiple times.

procedure

(plug context expression)  any

  context : any/c
  expression : any/c
The first argument to this function is an term to plug into. The second argument is the term to replace in the first argument. It returns the replaced term. This is also used when a term sub-expression contains in-hole.

procedure

(variable-not-in t prefix)  symbol?

  t : any/c
  prefix : symbol?
A helper function that accepts a term and a variable. It returns a symbol that not in the term, where the variable has prefix as a prefix.

procedure

(variables-not-in t vars)  (listof symbol?)

  t : any/c
  vars : (listof symbol?)
Like variable-not-in, create variables that do no occur in tbut returning a list of such variables, one for each variable in its second argument.

The variables-not-in function does not expect the symbols in vars to be distinct, but it does produce a list of distinct symbols.

procedure

(exn:fail:redex? v)  boolean?

  v : any/c
Returns #t if its argument is a Redex exception record, and #f otherwise.

4.3 Languages

syntax

(define-language lang-name
  non-terminal-def ...
  maybe-binding-spec)
 
non-terminal-def = (non-terminal-name ...+ ::= pattern ...+)
  | (non-terminal-name pattern ...+)
  | ((non-terminal-name ...+) pattern ...+)
     
maybe-binding-spec = 
  | #:binding-forms binding-pattern ...
     
binding-pattern = pattern
  | binding-pattern #:exports beta
  | binding-pattern #:refers-to beta
  | binding-pattern #:...bind (id beta beta)
     
beta = nothing
  | symbol
  | (shadow beta-sequence ...)
     
beta-sequence = beta
  | ... ; literal ellipsis
Defines the grammar of a language. The define-language form supports the definition of recursive patterns, much like a BNF, but for regular-tree grammars. It goes beyond their expressive power, however, because repeated name, in-hole, and side-condition patterns can restrict matches in complex ways.

A non-terminal-def comprises one or more non-terminal names (considered aliases) followed by one or more productions.

For example, the following defines lc-lang as the grammar of the λ-calculus:
(define-language lc-lang
  (e ::= (e e ...)
         x
         (λ (x ...) e))
  (v ::= (λ (x ...) e))
  (E ::= (v ... E e ...)
         hole)
  (x y ::= variable-not-otherwise-mentioned))

It has non-terminals: e for the expression language, x and y for variables, v for values, and E for the evaluation contexts.

Non-terminals used in define-language are not bound in side-condition patterns. Duplicate non-terminals that appear outside of the binding-forms section are not constrained to be the same unless they have underscores in them.

4.3.1 Binding Forms

Typical languages provide a mechanism for the programmer to introduce new names and give them meaning. The language forms used for this (such as Racket’s let and λ) are called binding forms.

Binding forms require special treatment from the language implementer. In Redex, this treatment consists of declaring the binding forms at the time of language definition. Explicitly declaring binding forms makes safely manipulating terms containing binding simpler and easier, eliminating the need to write operations that (explicitly) respect the binding structure of the language.

When maybe-binding-spec is provided, it declares binding specifications for certain forms in the language. The binding-pattern specification is an extension of Redex’s pattern language, allowing the keywords #:refers-to, #:exports, and #:...binds to appear nested inside a binding pattern.

The language, lc-lang, above does not declare any binding specifications, despite the clear intention of λ as a binding form. To understand the consequences of not specifying any binding forms, consider the behavior of substitution on terms of lc-lang.

Passing the #:lang argument to term allows the substitute metafunction to determine the language of its arguments.

> (term (substitute (x (λ (x) (λ (y) x)))
                    x
                    (y y)) #:lang lc-lang)

'((y y) (λ ((y y)) (λ (y) (y y))))

This call is intended to replace all free occurrences of x with (y y) in the first argument to substitute. But, because lc-lang is missing a binding forms declaration, substitute replaces all instances of x with (y y) in the term (x (λ (x) (λ (y) x))). Note that even the x that appears in what is normally a binding position has been replaced, resulting in an ill-formed lambda expression.

In order to have substitute behave correctly when substituting over terms that contain bound variables, the language lc-lang must declare its binding specification. Consider the following simplification of the lc-lang definition, this time with a binding form declaration for λ.

(define-language lc-bind
  (e ::= (e e)
         x
         (λ (x) e))
  (v ::= (λ (x) e))
  (x y ::= variable-not-otherwise-mentioned)
  #:binding-forms
  (λ (x) e #:refers-to x))

Just like Racket’s λ, in lc-bind all instances of the argument variable in the body of the lambda refer to the argument. In a binding declaration, this is specified using the #:refers-to keyword. Now the previous example has the right behavior.

> (term (substitute (x (λ (x) (λ (y) x)))
                    x
                    (y y)) #:lang lc-bind)

'((y y) (λ (x«0») (λ (y«1») x«0»)))

Note that sometimes substitute changes the names of the bound identifiers, in this case replacing the x and y with identifiers that have « and » in their names.

The #:refers-to declaration says that, in a λ term, the e subterm has the name from the x subterm in scope.

4.3.2 Multiple Variables in a Single Scope

To generalize to the version of λ in lc-lang, we need to cope with multiple variables at once. And in order to do that, we must handle the situation where some of the names are the same. Redex’s binding support offers only one option for this, namely taking the variables in order. The is captured by the keyword shadow. It also allows us to specify the binding structure for let:

(define-language lc-bind+let
  (e ::= x
         number
         (λ (x ...) e)
         (e e)
         (let ([x e] ...) e))
  (x ::= variable-not-otherwise-mentioned)
  #:binding-forms
  (λ (x ...) e #:refers-to (shadow x ...))
  (let ([x e_x] ...) e_body #:refers-to (shadow x ...)))

This #:binding-forms declaration says that the subterm e of the λ expression refers to all of the binders in λ. Similarly, the e_body refers to all of the binders in the let expression.

> (term (substitute (let ([x 5] [y x]) (y x))
                    x
                    z) #:lang lc-bind+let)

'(let ((x«2» 5) (y«3» z)) (y«3» x«2»))

The intuition behind the name of the shadow form can be seen in the following example:

> (term (substitute (let ([x 1] [y x] [x 3]) x)
                    x
                    z) #:lang lc-bind+let)

'(let ((x«4» 1) (y«5» z) (x«6» 3)) x«4»)

Because the lc-bind+let language does not require that all binders in its let form be distinct from one another, the binding forms specification must declare what happens when there is a conflict. The shadow form specifies that duplicate binders will be shadowed by earlier binders in its list of arguments. (Of course, if we were interested in modelling Racket’s let form, we’d want that term to be malformed syntax.)

It is possible to have multiple uses of #:refers-to in a single binding specification. For example, consider a language with a letrec form.

(define-language lc-bind+letrec
  (e ::= x
         number
         (λ (x ...) e)
         (e e)
         (let ([x e] ...) e)
         (letrec ([x e] ...) e))
  (x ::= variable-not-otherwise-mentioned)
  #:binding-forms
  (λ (x ...) e #:refers-to (shadow x ...))
  (let ([x e_x] ...) e_body #:refers-to (shadow x ...))
  (letrec ([x e_x] ...) #:refers-to (shadow x ...) e_body #:refers-to (shadow x ...)))

In this binding specification the subterms corresponding to both ([x e_x] ...) and e_body refer to the bound variables (shadow x ...).

> (term (substitute (letrec ([x x]) x) x y) #:lang lc-bind+letrec)

'(letrec ((x«7» x«7»)) x«7»)

> (term
   (substitute
    (letrec ([x (λ (a) (y a))]
             [y (λ (b) (z b))]
             [z a])
      (x 7))
    a
    (λ (x) 5))
   #:lang lc-bind+letrec)

'(letrec ((x«8» (λ (a«11») (y«9» a«11»)))

          (y«9» (λ (b«12») (z«10» b«12»)))

          (z«10» (λ (x) 5)))

   (x«8» 7))

4.3.3 Ellipses in Binding Forms

Some care must be taken when writing binding specifications that match patterns with ellipses. If a pattern symbol is matched underneath ellipses, it may only be mentioned under the same number of ellipses. Consider, for example, a language with Racket’s let-values binding form.

(define-language lc-bind+values
  (e ::= x
         number
         (λ (x ...) e)
         (e e)
         (values e ...)
         (let-values ([(x ...) e] ...) e))
  (x ::= variable-not-otherwise-mentioned)
  #:binding-forms
  (λ (x ...) e #:refers-to (shadow x ...))
  (let-values ([(x ...) e_x0] ...)
    e_body #:refers-to (shadow (shadow x ...) ...)))

In the binding specification for the let-values form, the bound variable, x, occurs only under a single ellipsis, thus when it is mentioned in a #:refers-to clause it is restricted to be mentioned only underneath a single ellipsis. Therefore the body of the let-values form must refer to (shadow (shadow x ...) ...) rather than (shadow x ... ...).

4.3.4 Compound Forms with Binders

So far, the nonterminals mentioned in #:refers-to have always stood directly for variables that appear in the terms. But sometimes the variables are down inside some piece of the term, or only some of the variables are relevant. The #:exports clause can be used to handle such situations.

When a binding form with an #:exports clause is mentioned, the names brought into scope are determined by recursively examining everything mentioned by that #:exports clause. Consider the following version of the lc-bind language with lists that allows for pattern matching in binding positions.

(define-language lc-bind+patterns
    (e ::= x
           number
           (λ (p) e)
           (e e)
           (list e ...))
    (x ::= variable-not-otherwise-mentioned)
    (p ::= (listp p ...) x)
    #:binding-forms
    (λ (p) e #:refers-to p)
    (listp p ...) #:exports (shadow p ...))

In this language functions accept patterns as arguments, therefore all variables mentioned in a pattern in binding position should be bound in the body of the function. A call to the substitute metafunction shows this behavior.

> (term
   (substitute (x (λ ((listp w (listp x y) z)) (list z y x w)))
               x
               u)
   #:lang lc-bind+patterns)

'(u

  (λ ((listp w«13» (listp x«14» y«15») z«16»)) (list z«16» y«15» x«14» w«13»)))

The use of the #:exports clause in the binding specification for lc-bind+patterns allows the use of nested binding patterns seen in the example. More precisely, each p may itself be a pattern that mentions any number of bound variables.

4.3.5 Binding Repetitions

In some situations, the #:exports and #:refers-to keywords are not sufficiently expressive to be able to describe the binding structure of different parts of a repeated sequence relate to each other. For example, consider the let* form. Its shape is the same as let, namely (let* ([x e] ...) e), but the binding structure is different.

In a let* form, each variable is accessible to each of the es that follow it, with all of the variables available in the body (the final e). With #:exports, we can build an expression form that has a structure like that, but we must write syntax that nests differently than let*.

(define-language lc-bind+awkward-let*
  (e ::= (let*-awk c e) natural x (+ e ...))
  (x ::= variable-not-otherwise-mentioned)
  (c ::= (clause x e c) ())
  #:binding-forms
  (let*-awk c e #:refers-to c)
  (clause x e c #:refers-to x) #:exports (shadow x c))

The let*-awk form binds like Racket’s let*, with each clause’s variable being active for the subsequent ones, but the syntax is different with extra nesting inside the clauses:

> (term (substitute (let*-awk (clause x y (clause y x ()))
                              (+ x y z))
                    x
                    1)
        #:lang lc-bind+awkward-let*)

'(let*-awk (clause x«17» y (clause y«18» x«17» ())) (+ x«17» y«18» z))

> (term (substitute (let*-awk (clause x y (clause y x ()))
                              (+ x y z))
                    y
                    2)
        #:lang lc-bind+awkward-let*)

'(let*-awk (clause x«19» 2 (clause y«20» x«19» ())) (+ x«19» y«20» z))

In order to get the same syntax as Racket’s let*, we need to use the #:...bind binding pattern annotation. A #:...bind can appear wherever a ... might appear, and it has the same function, namely indicating a repetition of the preceding pattern. In addition, however it comes with three extra pieces that follow the #:...bind form that describe how the binding structure inside the repetition is handled. The first part is a name that can be used by a #:refers-to outside of the repetition to indicate all of the exported variables of the sequence. The middle piece indicates the variables from a specific repetition of the ellipsis are exported to all subsequent repetitions of the ellipsis. The last piece is a beta that moves backwards through the sequence, indicating what is exported from the last repetition of the sequence to the one before, from the one before to the one before that, and then finally from the first one to the export of the entire sequence (as named by the identifier in the first position).

So, in this example, we use #:...bind to express the scope of let*.

(define-language lc-bind+let*
  (e ::= (let* ([x e] ...) e) natural x (+ e ...))
  (x ::= variable-not-otherwise-mentioned)
  #:binding-forms
  (let* ([x e] #:...bind (clauses x (shadow clauses x)))
    e_body #:refers-to clauses))

It says that the name of the exported variables from the entire sequence is clauses, which means that all of the variable exported from the sequence in the second position of the let* bind variables in the body (thanks to the last #:refers-to in the example). The x in the second position following the #:...bind says that x is in scope for each of the subsequent [x e] elements of the sequence. The final (shadow clauses x) says that the variables in a subsequent clauses are exported by the current one, as well as x, which then is exported by the entire sequence.

> (term (substitute (let* ([x y] [y x])
                      (+ x y z))
                    x
                    1)
        #:lang lc-bind+let*)

'(let* ((x«22» y) (y«21» x«22»)) (+ x«22» y«21» z))

> (term (substitute (let* ([x y] [y x])
                      (+ x y z))
                    y
                    2)
        #:lang lc-bind+let*)

'(let* ((x«24» 2) (y«23» x«24»)) (+ x«24» y«23» z))

syntax

::=

A non-terminal’s names and productions may be separated by the keyword ::=. Use of the ::= keyword outside a language definition is a syntax error.

syntax

shadow

Recognized specially within a define-language. A shadow is an error elsewhere.

syntax

nothing

Recognized specially within a define-language. A nothing is an error elsewhere.

syntax

(define-extended-language extended-lang base-lang
  non-terminal-def ...
  maybe-binding-spec)
 
non-terminal-def = (non-terminal-name ...+ ::= pattern ...+)
  | (non-terminal-name pattern ...+)
  | ((non-terminal-name ...+) pattern ...+)
     
maybe-binding-spec = 
  | #:binding-forms binding-declaration ...
Extends a language with some new, replaced, or extended non-terminals. For example, this language:

(define-extended-language lc-num-lang
  lc-lang
  (e ::= ....     ; extend the previous `e' non-terminal
     number
     +)
  (v ::= ....     ; extend the previous `v' non-terminal
     number
     +))

extends lc-lang with two new alternatives (+ and number) for the v non-terminal, carries forward the e, E, x, and y non-terminals. Note that the meaning of variable-not-otherwise-mentioned adapts to the language where it is used, so in this case it is equivalent to (variable-except λ +) because λ and + are used as literals in this language.

The four-period ellipses indicates that the new language’s non-terminal has all of the alternatives from the original language’s non-terminal, as well as any new ones. If a non-terminal occurs in both the base language and the extension, the extension’s non-terminal replaces the originals. If a non-terminal only occurs in the base language, then it is carried forward into the extension. And, of course, define-extended-language lets you add new non-terminals to the language.

If a language has a group of multiple non-terminals defined together, extending any one of those non-terminals extends all of them.

syntax

(define-union-language L base/prefix-lang ...)

 
base/prefix-lang = lang-id
  | (prefix lang-id)
Constructs a language that is the union of all of the languages listed in the base/prefix-lang.

If the two languages have non-terminals in common, then define-union-language will combine all of the productions of the common non-terminals. For example, this definition of L:
(define-language L1
  (e ::=
     (+ e e)
     number))
(define-language L2
  (e ::=
     (if e e e)
     true
     false))
(define-union-language L1-plus-L2 L1 L2)
is equivalent to this one:
(define-language L1-plus-L2
  (e ::=
     (+ e e)
     number
     (if e e e)
     true
     false))

If a language has a prefix, then all of the non-terminals from that language have the corresponding prefix in the union language. The prefix helps avoid unintended collisions between the constituent language’s non-terminals.

For example, with two these two languages:
(define-language UT
  (e (e e)
     (λ (x) e)
     x))
 
(define-language WT
  (e (e e)
     (λ (x t) e)
     x)
  (t ( t t)
     num))
then this declaration:

(define-union-language B (ut. UT) (wt. WT))

will create a language named B containing the non-terminals ut.e, wt.e, and wt.t consisting of the productions listed in the original languages.

procedure

(make-immutable-binding-hash lang [assocs])  dict?

  lang : compiled-lang?
  assocs : (listof pair?) = '()
Returns an immutable dictionary where alpha-equivalent? keys are treated as the same.

Added in version 1.14 of package redex-lib.

procedure

(make-binding-hash lang [assocs])  dict?

  lang : compiled-lang?
  assocs : (listof pair?) = '()
Returns a mutable dictionary where alpha-equivalent? keys are treated as the same.

Added in version 1.14 of package redex-lib.

procedure

(language-nts lang)  (listof symbol?)

  lang : compiled-lang?
Returns the list of non-terminals (as symbols) that are defined by this language.

procedure

(compiled-lang? l)  boolean?

  l : any/c
Returns #t if its argument was produced by language, #f otherwise.

parameter

(default-language)  (or/c false/c compiled-lang?)

(default-language lang)  void?
  lang : (or/c false/c compiled-lang?)
The value of this parameter is used by the default value of (default-equiv) to determine what language to calculate alpha-equivalence in. By default, it is #f, which acts as if it were a language with no binding forms. In that case, alpha-equivalence is the same thing as equal?.

The default-language parameter is set to the appropriate language inside judgment forms and metafunctions, and by apply-reduction-relation.

procedure

(alpha-equivalent? lang lhs rhs)  boolean?

  lang : compiled-lang?
  lhs : any/c
  rhs : any/c
(alpha-equivalent? lhs rhs)  boolean?
  lhs : any/c
  rhs : any/c
Returns #t if (according to the binding specification in lang) the bound names in lhs and rhs have the same structure and, in everything but bound names, they are equal?. If lang has no binding forms, terms have no bound names and therefore alpha-equivalent? is the same as equal?.

If the lang argument is not supplied, it defaults to the value of (default-language), which must not #f.

metafunction

(substitute val old-var new-val)

A metafunction that returns a value like val, except that any free occurences of old-var have been replaced with new-val, in a capture-avoiding fashion. The bound names of val may be freshened in order to accomplish this, based on the binding information in (default-language) (this is unlike normal metafunctions, which are defined in a particular language).

Note that substitute is merely a convenience metafunction. Any manually-written substitution in the correct language will also be capture-avoiding, provided that the language’s binding forms are correctly defined. However, substitute may be significantly faster.

4.4 Reduction Relations

syntax

(reduction-relation language domain codomain base-arrow
                    reduction-case ...
                    shortcuts)
 
domain = 
  | #:domain pattern
     
codomain = 
  | #:codomain pattern
     
base-arrow = 
  | #:arrow base-arrow-name
     
reduction-case = (arrow-name pattern term red-extras ...)
     
red-extras = rule-name
  | (fresh fresh-clause ...)
  | (side-condition racket-expression)
  | (where pattern term)
  | (where/hidden pattern term)
  | (where/error pattern term)
  | (bind pattern term)
  | (bind/hidden pattern term)
  | (judgment-holds (judgment-form-id pat/term ...))
  | (judgment-holds (relation-id term ...))
  | (side-condition/hidden racket-expression)
     
shortcuts = 
  | with shortcut ...
     
shortcut = 
[(old-arrow-name pattern term)
 (new-arrow-name identifier identifier)]
     
rule-name = identifier
  | string
  | (computed-name racket-expression)
     
fresh-clause = var
  | ((var1 ...) (var2 ...))
     
pat/term = pattern
  | term
Defines a reduction relation case-wise, one case for each of the reduction-case clauses.

The optional domain and codomain clauses provide contracts for the relation. If the codomain is not present, but the domain is, then the codomain is expected to be the same as the domain.

The arrow-name in each reduction-case clause is either base-arrow-name (default -->) or an arrow name defined by shortcuts (described below). In either case, the pattern refers to language and binds variables in the corresponding term. Following the pattern and term can be the name of the reduction rule and declarations of fresh variables and side-conditions.

For example, the expression

(reduction-relation
 lc-lang
 (--> (in-hole c_1 ((λ (variable_i ...) e_body) v_i ...))
      (in-hole c_1 ,(foldl lc-subst
                           (term e_body)
                           (term (v_i ...))
                           (term (variable_i ...))))
      beta-v))

defines a reduction relation for the lc-lang grammar.

A rule’s name (used in typesetting, the stepper, traces, and apply-reduction-relation/tag-with-names) can be given as a literal (an identifier or a string) or as an expression that computes a name using the values of the bound pattern variables (much like the rule’s right-hand side). Some operations require literal names, so a rule definition may provide both a literal name and a computed name. In particular, only rules that include a literal name may be replaced using extend-reduction-relation, used as breakpoints in the stepper, and selected using render-reduction-relation-rules. The output of apply-reduction-relation/tag-with-names, traces, and the stepper prefers the computed name, if it exists. Typesetting a rule with a computed name shows the expression that computes the name only when the rule has no literal name or when it would not typeset in pink due to with-unquote-rewriters in the context; otherwise, the literal name (or nothing) is shown.

Fresh variable clauses generate variables that do not occur in the term being reduced. If the fresh-clause is a variable, that variable is used both as a binding in the term and as the prefix for the freshly generated variable. (The variable does not have to be a non-terminal in the language of the reduction relation.)

The second form of fresh-clauses generates a sequence of variables. In that case, the ellipses are literal ellipses; that is, you must actually write ellipses in your rule. The variable var1 is like the variable in first case of a fresh-clause; namely it is used to determine the prefix of the generated variables and it is bound in the right-hand side of the reduction rule, but unlike the single-variable fresh clause, it is bound to a sequence of variables. The variable var2 is used to determine the number of variables generated and var2 must be bound by the left-hand side of the rule.

The expressions within side-condition clauses and side-condition/hidden clauses are collected with and and used as guards on the case being matched. The argument to each side-condition should be a Racket expression, and the pattern variables in the pattern are bound in that expression. A side-condition/hidden clause is the same as a side-condition clause, except that the condition is not rendered when typesetting via redex/pict.

Each where clause acts as a side condition requiring a successful pattern match, and it can bind pattern variables in the side-conditions (and where clauses) that follow and in the metafunction result.

A where/hidden clause is the same as a where clause, but the clause is not rendered when typesetting via redex/pict.

The where/error clause clause is like where, except that a failure to match is an error and, if multiple matches are possible, the right-hand side must produce the same result for each of the different matches (in the sense of alpha-equivalent? using the language that this reduction relation is defined with).

Each judgment-holds clause acts like a where clause, where the left-hand side pattern incorporates each of the patterns used in the judgment form’s output positions.

Each shortcut clause defines arrow names in terms of base-arrow-name and earlier shortcut definitions. The left- and right-hand sides of a shortcut definition are identifiers, not patterns and terms. These identifiers need not correspond to non-terminals in language and if they do, that correspondence is ignored (more precisely, the shortcut is not restricted only to terms matching the non-terminal).

For example, this expression

(reduction-relation
 lc-num-lang
 (==> ((λ (variable_i ...) e_body) v_i ...)
      ,(foldl lc-subst
              (term e_body)
              (term (v_i ...))
              (term (variable_i ...))))
 (==> (+ number_1 ...)
      ,(apply + (term (number_1 ...))))
 
 with
 [(--> (in-hole c_1 a) (in-hole c_1 b))
  (==> a b)])

defines reductions for the λ-calculus with numbers, where the ==> shortcut is defined by reducing in the context c.

A fresh clause in reduction-case defined by shortcut refers to the entire term, not just the portion matched by the left-hand side of shortcut’s use.

syntax

(extend-reduction-relation reduction-relation language more ...)

This form extends the reduction relation in its first argument with the rules specified in more. They should have the same shape as the rules (including the with clause) in an ordinary reduction-relation.

If the original reduction-relation has a rule with the same name as one of the rules specified in the extension, the old rule is removed.

In addition to adding the rules specified to the existing relation, this form also reinterprets the rules in the original reduction, using the new language.
Combines all of the argument reduction relations into a single reduction relation that steps when any of the arguments would have stepped.

Returns the names of the reduction relation’s named clauses.

syntax

(compatible-closure reduction-relation lang non-terminal)

This accepts a reduction, a language, the name of a non-terminal in the language and returns the compatible closure of the reduction for the specified non-terminal.

syntax

(context-closure reduction-relation lang pattern)

This accepts a reduction, a language, a pattern representing a context (i.e., that can be used as the first argument to in-hole; often just a non-terminal) in the language and returns the closure of the reduction in that context.

procedure

(reduction-relation? v)  boolean?

  v : any/c
Returns #t if its argument is a reduction-relation, and #f otherwise.

procedure

(apply-reduction-relation r t)  (listof any/c)

  r : (or/c reduction-relation? IO-judgment-form?)
  t : any/c
This accepts reduction relation, a term, and returns a list of terms that the term reduces to.

Like apply-reduction-relation, but the result indicates the names of the reductions that were used.

procedure

(apply-reduction-relation* r 
  t 
  [#:all? all 
  #:cache-all? cache-all? 
  #:stop-when stop-when]) 
  (listof any/c)
  r : (or/c reduction-relation? IO-judgment-form?)
  t : any/c
  all : boolean? = #f
  cache-all? : boolean? = (or all? (current-cache-all?))
  stop-when : (-> any/c any) = (λ (x) #f)
Accepts a reduction relation and a term. Starting from t, it follows every reduction path and returns either all of the terms that do not reduce further or all of the terms, if all? is #t. If there are infinite reduction sequences that do not repeat, this function will not terminate (it does terminate if the only infinite reduction paths are cyclic).

If the cache-all? argument is #t, then apply-reduction-relation* keeps a cache of all visited terms when traversing the graph and does not revisit any of them. This cache can, in some cases, use a lot of memory, so it is off by default and the cycle checking happens by keeping track only of the current path it is traversing through the reduction graph.

The stop-when argument controls the stopping criterion. Specifically, it is called with each term that apply-reduction-relation* encounters. If it ever returns a true value (anything except #f), then apply-reduction-relation* considers the term to be irreducible (and so returns it and does not try to reduce it further).

parameter

(current-cache-all?)  boolean?

(current-cache-all? cache-all?)  void?
  cache-all? : boolean?
Controls the behavior of apply-reduction-relation* and test-->>’s cycle checking. See apply-reduction-relation* for more details.

Examples:
> (define-language empty-lang)
> (define R
    (reduction-relation
     empty-lang
     (--> 0 1)
     (--> 0 2)
     (--> 2 3)
     (--> 3 3)))
> (apply-reduction-relation R 0)

'(2 1)

> (apply-reduction-relation* R 0)

'(1)

syntax

-->

Recognized specially within reduction-relation. A --> form is an error elsewhere.

syntax

fresh

Recognized specially within reduction-relation. A fresh form is an error elsewhere.

syntax

with

Recognized specially within reduction-relation. A with form is an error elsewhere.

4.5 Other Relations

syntax

(define-metafunction language
  metafunction-contract
  [(name pattern ...) term metafunction-extras ...]
  ...)
 
metafunction-contract = 
  | 
id : pattern-sequence ... -> range
maybe-pre-condition
maybe-post-condition
     
maybe-pre-condition = #:pre term
  | 
     
maybe-post-condition = #:post term
  | 
     
range = pattern
  | pattern or range
  | pattern  range
  | pattern  range
     
metafunction-extras = (side-condition racket-expression)
  | (side-condition/hidden racket-expression)
  | (where pat term)
  | (where/hidden pat term)
  | (where/error pat term)
  | 
(judgment-holds
 (judgment-form-id pat/term ...))
  | 
(judgment-holds
 (relation-id term ...))
  | (clause-name name)
  | or term
A metafunction is a function on terms. The define-metafunction form builds a metafunction 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 contract, if present, is matched against every input to the metafunction and, if the match fails, an exception is raised. If a metavariable is repeated in a contract, it does not require the terms to be equal, unless there is an underscore subscript (i.e., the binding works like it does in define-language, not how it works in the patterns in the left-hand sides of the metafunction clauses).

If present, the term inside the maybe-pre-condition is evaluated after a successful match to the input pattern in the contract (with any variables from the input contract bound). If it returns #f, then the input contract is considered to not have matched and an error is also raised. When a metafunction returns, the expression in the maybe-post-condition is evaluated (if present), with any variables from the input or output contract bound.

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

The resulting metafunction raises an exception recognized by exn:fail:redex? if no clauses match or if one of the clauses matches multiple ways (and that leads to different results for the different matches).

The side-condition extra is evaluated after a successful match to the corresponding argument pattern. If it returns #f, the clause is considered not to have matched, and the next one is tried. The side-condition/hidden extra behaves the same, but is not typeset.

The where and where/hidden extra are like side-condition and side-condition/hidden, except the match guards the clause. The where/error extra is like where, except that the pattern must match.

The judgment-holds clause is like side-condition and where, except the given judgment or relation must hold for the clause to be taken.

The clause-name is used only when typesetting. See metafunction-cases.

The or clause is used to define piecewise conditional metafunctions. In particular, if any of the where or side-condition clauses fail, then evaluation continues after an or clause, treating the term that follows as the result (subject to any subsequent where clauses or side-conditions. This construction is equivalent to simply duplicating the left-hand side of the clause, once for each or expression, but signals to the typesetting library to use a large left curly brace to group the conditions in the or.

For example, here are two equivalent definitions of a biggest metafunction that typeset differently:

Examples:
> (define-metafunction lc-lang
    biggest : natural natural -> natural
    [(biggest natural_1 natural_2)
     natural_2
     (side-condition (< (term natural_1) (term natural_2)))]
    [(biggest natural_1 natural_2)
     natural_1])
> (render-metafunction biggest)

image

> (define-metafunction lc-lang
    biggest : natural natural -> natural
    [(biggest natural_1 natural_2)
     natural_2
     (side-condition (< (term natural_1) (term natural_2)))
  
     or
  
     natural_1])
> (render-metafunction biggest)

image

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 (λ (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 λ 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.

Changed in version 1.4 of package redex-lib: Added #:post conditions.
Changed in version 1.5: Added or clauses.

syntax

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

syntax

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

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

syntax

(define-judgment-form language
  mode-spec
  contract-spec
  invariant-spec
  rule rule ...)
 
mode-spec = #:mode (form-id pos-use ...)
     
contract-spec = 
  | #:contract (form-id pattern-sequence ...)
     
invariant-spec = #:inv term
  | 
     
pos-use = I
  | O
     
rule = 
[premise
 ...
 dashes rule-name
 conclusion]
  | 
[conclusion
 premise
 ...
 rule-name]
     
conclusion = (form-id pat/term ...)
     
premise = (judgment-form-id pat/term ...) maybe-ellipsis
  | (relation-id pat/term ...) maybe-ellipsis
  | (where pattern term)
  | (where/hidden pattern term)
  | (where/error pattern term)
  | (side-condition term)
  | (side-condition/hidden term)
     
rule-name = 
  | string
  | non-ellipsis-non-dashes-var
     
pat/term = pattern
  | term
     
maybe-ellipsis = 
  | ...
     
dashes = ---
  | ----
  | -----
  | etc.
Defines form-id as a relation on terms via a set of inference rules. Each rule must be such that its premises can be evaluated left-to-right without “guessing” values for any of their pattern variables. Redex checks this property using the mandatory mode-spec declaration, which partitions positions into inputs I and outputs O. Output positions in conclusions and input positions in premises must be terms; input positions in conclusions and output positions in premises must be patterns. The rule-names are used by build-derivations and by render-judgment-form.

When the optional contract-spec declaration is present, Redex dynamically checks that the terms flowing through these positions match the provided patterns, raising an exception recognized by exn:fail:redex? if not. The term in the optional invariant-spec is evaluated after the output positions have been computed and the contract has matched successfully, with variables from the contract bound; a result of #f is considered to be a contract violation and an exception is raised.

For example, the following defines addition on natural numbers:
> (define-language nats
    (n ::= z (s n)))
> (define-judgment-form nats
    #:mode (sum I I O)
    #:contract (sum n n n)
    [-----------  "zero"
     (sum z n n)]
  
    [(sum n_1 n_2 n_3)
     ------------------------- "add1"
     (sum (s n_1) n_2 (s n_3))])

The judgment-holds form checks whether a relation holds for any assignment of pattern variables in output positions.

Examples:
> (judgment-holds (sum (s (s z)) (s z) (s (s (s z)))))

#t

> (judgment-holds (sum (s (s z)) (s z) (s (s (s n)))))

#t

> (judgment-holds (sum (s (s z)) (s z) (s (s (s (s n))))))

#f

Alternatively, this form constructs a list of terms based on the satisfying pattern variable assignments.

Examples:
> (judgment-holds (sum (s (s z)) (s z) (s (s (s n)))) n)

'(z)

> (judgment-holds (sum (s (s z)) (s z) (s (s (s (s n))))) n)

'()

> (judgment-holds (sum (s (s z)) (s z) (s (s (s n)))) (s n))

'((s z))

Declaring different modes for the same inference rules enables different forms of computation. For example, the following mode allows judgment-holds to compute all pairs with a given sum.
> (define-judgment-form nats
    #:mode (sumr O O I)
    #:contract (sumr n n n)
    [------------ sumr-z
     (sumr z n n)]
  
    [(sumr n_1 n_2 n_3)
     -------------------------- sumr-s
     (sumr (s n_1) n_2 (s n_3))])
> (judgment-holds (sumr n_1 n_2 (s (s z))) (n_1 n_2))

'(((s (s z)) z) ((s z) (s z)) (z (s (s z))))

A rule’s where, where/hidden, and where/error premises behave as in reduction-relation and define-metafunction.

Examples:
> (define-judgment-form nats
    #:mode (le I I)
    #:contract (le n n)
    [--------
     (le z n)]
  
    [(le n_1 n_2)
     --------------------
     (le (s n_1) (s n_2))])
> (define-metafunction nats
    pred : n -> n or #f
    [(pred z) #f]
    [(pred (s n)) n])
> (define-judgment-form nats
    #:mode (gt I I)
    #:contract (gt n n)
    [(where n_3 (pred n_1))
     (le n_2 n_3)
     ----------------------
     (gt n_1 n_2)])
> (judgment-holds (gt (s (s z)) (s z)))

#t

> (judgment-holds (gt (s z) (s z)))

#f

A rule’s side-condition and side-condition/hidden premises are similar to those in reduction-relation and define-metafunction, except that they do not implicitly unquote their right-hand sides. In other words, a premise of the form (side-condition term) is close to the premise (where #t term), except it does not typeset with the “#t = ”, as that would and it holds whenever the expression evaluates to any non #f value (not just #t).

Judgments with exclusively I mode positions may also be used in terms in a manner similar to metafunctions, and evaluate to a boolean.

Examples:
> (term (le (s z) (s (s z))))

#t

> (term (le (s z) z))

#f

A literal ellipsis may follow a judgment premise when a template in one of the judgment’s input positions contains a pattern variable bound at ellipsis-depth one.

Examples:
> (define-judgment-form nats
    #:mode (even I)
    #:contract (even n)
  
    [-------- "evenz"
     (even z)]
  
    [(even n)
     ---------------- "even2"
     (even (s (s n)))])
> (define-judgment-form nats
    #:mode (all-even I)
    #:contract (all-even (n ...))
    [(even n) ...
     ------------------
     (all-even (n ...))])
> (judgment-holds (all-even (z (s (s z)) z)))

#t

> (judgment-holds (all-even (z (s (s z)) (s z))))

#f

Redex evaluates premises depth-first, even when it doing so leads to non-termination. For example, consider the following definitions:
> (define-language vertices
    (v a b c))
> (define-judgment-form vertices
    #:mode (edge I O)
    #:contract (edge v v)
    [(edge a b)]
    [(edge b c)])
> (define-judgment-form vertices
    #:mode (path I I)
    #:contract (path v v)
    [----------
     (path v v)]
  
    [(path v_2 v_1)
     --------------
     (path v_1 v_2)]
  
    [(edge v_1 v_2)
     (path v_2 v_3)
     --------------
     (path v_1 v_3)])
Due to the second path rule, the follow query fails to terminate:

> (judgment-holds (path a c))

There are three example files that come with Redex that demonstrates three use cases.
  • "typing-rules.rkt" defines a type system in a way that supports mechanized typesetting. When a typing judgment form can be given a mode, it can also be encoded as a metafunction using where clauses as premises, but Redex cannot typeset that encoding as inference rules.

  • "sos.rkt" defines an SOS-style semantics in a way that supports mechanized typesetting.

  • "multi-val.rkt" defines a judgment form that serves as a multi-valued metafunction.

These files can be found via DrRacket’s File|Open Require Path... menu item. Type redex/examples/d/ into the dialog and then choose one of the names listed above. Or, evaluate the expression
(collection-file-path «filename.rkt»
                      "redex"
                      "examples"
                      "define-judgment-form")
replacing «filename.rkt» with one of the names listed above.

syntax

(define-extended-judgment-form language judgment-form-id
  mode-spec
  contract-spec
  invariant-spec
  rule ...)
Defines a new judgment form that extends judgment-form-id with additional rules. The mode-spec, contract-spec, invariant-spec, and rules are as in define-judgment-form.

The mode specification in this judgment form and the original must be the same.

syntax

(judgment-holds judgment-or-relation)

(judgment-holds judgment-or-relation term)
 
judgment-or-relation = (judgment-form-id pat/term ...)
  | (relation-id pat/term ...)
In its first form, checks whether judgment-or-relation holds for any assignment of the pattern variables in judgment-form-id’s output positions (or just that it holds in the case that a relation from define-relation is used). In its second form, produces a list of terms by instantiating the supplied term template with each satisfying assignment of pattern variables. In the second case, if a relation is supplied, there are no pattern variables, so the result is either a list with one element or the empty list.

> (judgment-holds (sum (s (s z)) (s z) n))

#t

> (judgment-holds (sum (s (s z)) (s z) n) n)

'((s (s (s z))))

See define-judgment-form for more examples.

syntax

(build-derivations judgment-or-relation)

Constructs all of the derivation trees for judgment-or-relation.

Example:
> (build-derivations (even (s (s z))))

(list

 (derivation

  '(even (s (s z)))

  "even2"

  (list (derivation '(even z) "evenz" '()))))

struct

(struct derivation (term name subs)
    #:extra-constructor-name make-derivation)
  term : any/c
  name : (or/c string? #f)
  subs : (listof derivation?)
Represents a derivation from a judgment form.

The term field holds an s-expression based rendering of the conclusion of the derivation, the name field holds the name of the clause with term as the conclusion, and subs contains the sub-derivations.

See also build-derivations.

syntax

I

Recognized specially within define-judgment-form, the I keyword is an error elsewhere.

syntax

O

Recognized specially within define-judgment-form, the O keyword is an error elsewhere.

syntax

(define-relation language
  relation-contract
  [(name pattern ...)
   term ...
   metafunction-extras ...] ...)
 
relation-contract = 
  | form-id  pattern x ... x pattern
  | form-id  pattern × ... × pattern
Similar to define-judgment-form but suitable only when every position is an input. Querying the result uses judgment-holds or the same syntax as metafunction application.

The contract specification for a relation restricts the patterns that can be used as input to a relation. For each argument to the relation, there should be a single pattern, using x or × to separate the argument contracts.

Examples:
> (define-language types
    ((τ σ) int
           num
           (τ  τ)))
> (define-relation types
    subtype  τ × τ
    [(subtype int num)]
    [(subtype (τ_1  τ_2) (σ_1  σ_2))
     (subtype σ_1 τ_1)
     (subtype τ_2 σ_2)]
    [(subtype τ τ)])
> (judgment-holds (subtype int num))

#t

> (judgment-holds (subtype (int  int) (num  num)))

#f

> (judgment-holds (subtype (num  int) (num  num)))

#t

procedure

(judgment-form? v)  boolean?

  v : any/c
Identifies values bound to identifiers introduced by define-judgment-form and define-relation.

procedure

(IO-judgment-form? v)  boolean?

  v : any/c
Identifies values bound to identifiers introduced by define-judgment-form when the mode is (I O) or (O I).

parameter

(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.

The tracing looks just like the tracing done by the racket/trace library, except that the first column printed by each traced call indicate if this call to the metafunction is cached. Specifically, a c is printed in the first column if the result is just returned from the cache and a space is printed if the metafunction call is actually performed.

Defaults to '().

4.6 Testing

syntax

(test-equal e1 e2 option)

 
option = #:equiv pred-expr
  | 
 
  pred-expr : (-> any/c any/c any/c)
Tests to see if e1 is equal to e2, using pred-expr as the comparison. It defaults to (default-equiv).

Examples:
> (define-language L
    (bt ::=
        empty
        (node any bt bt))
    (lt ::=
        empty
        (node any lt empty)))
> (define-metafunction L
    linearize/a : bt lt -> lt
    [(linearize/a empty lt) lt]
    [(linearize/a (node any_val bt_left bt_right) lt)
     (node any_val (linearize/a bt_left (linearize/a bt_right lt)) empty)])
> (define-metafunction L
    linearize : bt -> lt
    [(linearize bt) (linearize/a bt empty)])
> (test-equal (term (linearize empty))
              (term empty))
> (test-equal (term (linearize (node 1
                                     (node 2 empty empty)
                                     (node 3 empty empty))))
              (term (node 1 (node 2 (node 3 empty empty) empty) empty)))
> (test-results)

Both tests passed.

syntax

(test-->> rel-expr option ... e1-expr e2-expr ...)

 
option = #:cycles-ok
  | #:equiv pred-expr
  | #:pred pred-expr
 
  rel-expr : (or/c reduction-relation? IO-judgment-form?)
  pred-expr : (-> any/c any)
  e1-expr : any/c
  e2-expr : any/c
Tests to see if the term e1-expr, reduces to the terms e2-expr under rel-expr, using pred-expr to determine equivalence.

If #:pred is specified, it is applied to each reachable term until one of the terms fails to satisfy the predicate (i.e., the predicate returns #f). If that happens, then the test fails and a message is printed with the term that failed to satisfy the predicate.

The procedure supplied after #:equiv is always passed the result of reducing the expression as its first argument and (one of) the expected result(s) as its second argument.

This test uses apply-reduction-relation*, so it does not terminate when the resulting reduction graph is infinite, although it does terminate if there are cycles in the (finite) graph.

If #:cycles-ok is not supplied then any cycles detected are treated as a test failure. If a pred-expr is supplied, then it is used to compare the expected and actual results. If it isn’t supplied, then (default-equiv) is used.

syntax

(test--> rel-expr option ... e1-expr e2-expr ...)

 
option = #:equiv pred-expr
 
  rel-expr : (or/c reduction-relation? IO-judgment-form?)
  pred-expr : (-> any/c any/c any/c)
  e1-expr : any/c
  e2-expr : any/c
Tests to see if the term e1-expr, reduces to the terms e2-expr in a single rel-expr step, using pred-expr to determine equivalence (or (default-equiv) if pred-expr isn’t specified).

Examples:
> (define-language L
    (i integer))
> (define R
    (reduction-relation
     L
     (--> i i)
     (--> i ,(add1 (term i)))))
> (define (mod2=? i j)
    (= (modulo i 2) (modulo j 2)))
> (test--> R #:equiv mod2=? 7 1)

FAILED :10.0

expected: 1

  actual: 8

  actual: 7

> (test--> R #:equiv mod2=? 7 1 0)
> (test-results)

1 test failed (out of 2 total).

syntax

(test-->>∃ option ... rel-expr start-expr goal-expr)

 
option = #:steps steps-expr
 
  rel-expr : (or/c reduction-relation? IO-judgment-form?)
  start-expr : any/c
  goal-expr : 
(or/c (-> any/c any/c)
      (not/c procedure?))
  steps-expr : (or/c natural-number/c +inf.0)
Tests to see if the term start-expr reduces according to the reduction relation rel-expr to a term specified by goal-expr in steps-expr or fewer steps (default 1,000). The specification goal-expr may be either a predicate on terms or a term itself.

syntax

test-->>E

An alias for test-->>∃.

Examples:
> (define-language L
    (n natural))
> (define succ-mod8
    (reduction-relation
     L
     (--> n ,(modulo (add1 (term n)) 8))))
> (test-->>∃ succ-mod8 6 2)
> (test-->>∃ succ-mod8 6 even?)
> (test-->>∃ succ-mod8 6 8)

FAILED :17.0

term 8 not reachable from 6

> (test-->>∃ #:steps 6 succ-mod8 6 5)

FAILED :18.0

term 5 not reachable from 6 (within 6 steps)

> (test-results)

2 tests failed (out of 4 total).

syntax

(test-judgment-holds (judgment-form-or-relation pat/term ...))

Tests to see if (judgment-form-or-relation pat/term ...) holds.

syntax

(test-predicate p? e)

Tests to see if the value of e matches the predicate p?.

procedure

(test-results)  void?

Prints out how many tests passed and failed, and resets the counters so that next time this function is called, it prints the test results for the next round of tests.

parameter

(default-equiv)  (-> any/c any/c any/c)

(default-equiv equiv)  void?
  equiv : (-> any/c any/c any/c)
The value of this parameter is used as the default value of the equivalence predicates for test-equal, test-->, and test-->>.

It defaults to (lambda (lhs rhs) (alpha-equivalent? (default-language) lhs rhs)).

syntax

(make-coverage subject)

 
subject = metafunction
  | relation-expr
Constructs a structure (recognized by coverage?) to contain per-case test coverage of the supplied metafunction or reduction relation. Use with relation-coverage and covered-cases.

procedure

(coverage? v)  boolean?

  v : any/c
Returns #t for a value produced by make-coverage and #f for any other.

parameter

(relation-coverage)  (listof coverage?)

(relation-coverage tracked)  void?
  tracked : (listof coverage?)
Redex populates the coverage records in tracked (default null), counting the times that tests exercise each case of the associated metafunction and relations.

procedure

(covered-cases c)  (listof (cons/c string? natural-number/c))

  c : coverage?
Extracts the coverage information recorded in c, producing an association list mapping names (or source locations, in the case of metafunctions or unnamed reduction-relation cases) to application counts.

Examples:
> (define-language empty-lang)
> (define-metafunction empty-lang
    [(plus number_1 number_2)
     ,(+ (term number_1) (term number_2))])
> (define equals
    (reduction-relation
     empty-lang
     (--> (+) 0 "zero")
     (--> (+ number) number)
     (--> (+ number_1 number_2 number ...)
          (+ (plus number_1 number_2)
             number ...)
          "add")))
> (let ([equals-coverage (make-coverage equals)]
        [plus-coverage (make-coverage plus)])
    (parameterize ([relation-coverage (list equals-coverage
                                            plus-coverage)])
      (apply-reduction-relation* equals (term (+ 1 2 3)))
      (values (covered-cases equals-coverage)
              (covered-cases plus-coverage))))

'(("#f:22:0" . 1) ("add" . 2) ("zero" . 0))

'(("#f:21:0" . 2))

syntax

(generate-term from-pattern)

(generate-term from-judgment-form)
(generate-term from-metafunction)
(generate-term from-reduction-relation)
 
from-pattern = language pattern size-expr kw-args ...
  | language pattern
  | language pattern #:i-th index-expr
  | language pattern #:i-th
     
from-judgment-form = 
language #:satisfying
(judgment-form-id pattern ...)
  | 
language #:satisfying
(judgment-form-id pattern ...)
size-expr
     
from-metafunction = 
language #:satisfying
(metafunction-id pattern ...) = pattern
  | 
language #:satisfying
(metafunction-id pattern ...) = pattern
size-expr
  | #:source metafunction size-expr kw-args
  | #:source metafunction
     
from-reduction-relation = 
#:source reduction-relation-expr
size-expr kw-args ...
  | #:source reduction-relation-expr
     
kw-args = #:attempt-num attempts-expr
  | #:retries retries-expr
 
  size-expr : natural-number/c
  attempt-num-expr : natural-number/c
  retries-expr : natural-number/c
Generates terms in a number of different ways:
  • from-pattern: In the first case, randomly makes an expression matching the given pattern whose size is bounded by size-expr; the second returns a function that accepts a size bound and returns a random term. Calling this function (even with the same size bound) may be more efficient than using the first case.

    Examples:
    > (define-language L
        (e ::=
           (e e)
           (λ (x) e)
           x)
        (x ::= a b c))
    > (for/list ([i (in-range 10)])
        (generate-term L e 3))

    '((a (λ (c) b))

      ((a (λ (c) c)) a)

      a

      c

      ((λ (c) (b b)) (λ (a) (b c)))

      c

      (λ (b) (λ (b) a))

      c

      (c a)

      (c ((c a) a)))

    The #:i-th option uses an enumeration of the non-terminals in a language. If index-expr is supplied, generate-term returns the corresponding term and if it isn’t, generate-term returns a function from indices to terms.

    Example:
    > (for/list ([i (in-range 9)])
        (generate-term L e #:i-th i))

    '(a (a a) (λ (a) a) b (a (a a)) (λ (b) a) c ((a a) a) (λ (c) a))

    Base type enumerations such as boolean, natural and integer are what you might expect:

    Examples:
    > (for/list ([i (in-range 10)])
        (generate-term L boolean #:i-th i))

    '(#t #f #t #f #t #f #t #f #t #f)

    > (for/list ([i (in-range 10)])
        (generate-term L natural #:i-th i))

    '(0 1 2 3 4 5 6 7 8 9)

    > (for/list ([i (in-range 10)])
        (generate-term L integer #:i-th i))

    '(0 1 -1 2 -2 3 -3 4 -4 5)

    The real base type enumeration consists of all integers and flonums, and the number pattern consists of complex numbers with real and imaginary parts taken from the real enumeration.

    Examples:
    > (for/list ([i (in-range 20)])
        (generate-term L real #:i-th i))

    '(0

      +inf.0

      1

      -inf.0

      -1

      +nan.0

      2

      0.0

      -2

      4.9406564584125e-324

      3

      -4.9406564584125e-324

      -3

      9.8813129168249e-324

      4

      -9.8813129168249e-324

      -4

      1.4821969375237e-323

      5

      -1.4821969375237e-323)

    > (for/list ([i (in-range 20)])
        (generate-term L number #:i-th i))

    '(+inf.0

      0

      +inf.0+inf.0i

      1

      0+inf.0i

      -inf.0+inf.0i

      -inf.0

      0+1i

      +nan.0+inf.0i

      -1

      0-inf.0i

      0.0+inf.0i

      +nan.0

      0-1i

      4.9406564584125e-324+inf.0i

      2

      0+nan.0i

      -4.9406564584125e-324+inf.0i

      0.0

      0+2i)

    The string enumeration produces all single character strings before going on to strings with multiple characters. For each character it starts the lowercase Latin characters, then uppercase Latin, and then every remaining Unicode character. The variable enumeration is the same, except it produces symbols instead of strings.

    Examples:
    > (generate-term L string #:i-th 0)

    "a"

    > (generate-term L string #:i-th 1)

    "b"

    > (generate-term L string #:i-th 26)

    "A"

    > (generate-term L string #:i-th 27)

    "B"

    > (generate-term L string #:i-th 52)

    "\u0000"

    > (generate-term L string #:i-th 53)

    "\u0001"

    > (generate-term L string #:i-th 956)

    "μ"

    > (generate-term L variable #:i-th 1)

    'b

    > (generate-term L variable #:i-th 27)

    'B

    The variable-prefix, variable-except, and variable-not-otherwise-mentioned are defined similarly, as you expect.

    Examples:
    > (define-language L
        (used ::= a b c)
        (except ::= (variable-except a))
        (unused ::= variable-not-otherwise-mentioned))
    > (for/list ([i (in-range 10)])
        (generate-term L (variable-prefix a:) #:i-th i))

    '(a:a a:b a:c a:d a:e a:f a:g a:h a:i a:j)

    > (for/list ([i (in-range 10)])
        (generate-term L except #:i-th i))

    '(b c d e f g h i j k)

    > (for/list ([i (in-range 10)])
        (generate-term L unused #:i-th i))

    '(d e f g h i j k l m)

    Finally, the any pattern enumerates terms of the above base types.

    Example:
    > (for/list ([i (in-range 20)])
        (generate-term L any #:i-th i))

    '(()

      (())

      +inf.0

      (() ())

      "a"

      ((()))

      #t

      ((()) ())

      a

      (() . +inf.0)

      0

      ((()) . +inf.0)

      "b"

      (+inf.0)

      #f

      (+inf.0 ())

      b

      (+inf.0 . +inf.0)

      +inf.0+inf.0i

      (() () ()))

    In addition, all other pattern types are supported except for mismatch repeat ..._!_ patterns and side-condition patterns.

    The enumerators do not repeat terms unless the given pattern is ambiguous. Roughly speaking, the enumerator generates all possible ways that a pattern might be parsed and since ambiguous patterns have multiple ways they might be parsed, those multiple parsings turn into repeated elements in the enumeration.

    Example:
    > (for/list ([i (in-range 9)])
        (generate-term L (boolean_1 ... boolean_2 ...) #:i-th i))

    '(() (#t) (#t) (#t #t) (#f) (#t #f) (#f) (#f #t) (#f #f))

    Other sources of ambiguity are in-hole and overlapping non-terminals.

    Examples:
    > (define-language L
        (e ::= (e e) (λ (x) e) x)
        (E ::= hole (e E) (E e))
        (x ::= a b c))
    > (for/list ([i (in-range 9)])
        (generate-term L (in-hole E e) #:i-th i))

    '(a

      (a a)

      (a a)

      (a (a a))

      (λ (a) a)

      (a (λ (a) a))

      (a a)

      ((a a) a)

      ((λ (a) a) a))

    > (define-language L
        (overlap ::= natural integer))
    > (for/list ([i (in-range 10)])
        (generate-term L overlap #:i-th i))

    '(0 0 1 1 2 -1 3 2 4 -2)

    For similar reasons, enumerations for mismatch patterns (using _!_) do not work properly when given ambiguous patterns; they may repeat elements of the enumeration.

    Examples:
    > (define-language Bad
        (ambig ::= (x ... x ...)))
    > (generate-term Bad (ambig_!_1 ambig_!_1) #:i-th 4)

    '(() (x x))

    In this case, the elements of the resulting list are the same, even though they should not be, according to the pattern. Internally, the enumerator has discovered two different ways to generate ambig (one where the x comes from the first ellipses and one from the second) but those two different ways produce the same term and so the enumerator incorrectly produces (x x).

    See also redex-enum.

  • from-judgment-form: Randomly picks a term that satisfies the given use of the judgment form.

    Examples:
    > (define-language L
        (nat ::= Z (S nat)))
    > (define-judgment-form L
        #:mode (sum I I O)
        [---------------
         (sum Z nat nat)]
        [(sum nat_1 nat_2 nat_3)
         -------------------------------
         (sum (S nat_1) nat_2 (S nat_3))])
    > (for/list ([i (in-range 10)])
        (generate-term L #:satisfying
                       (sum nat_1 nat_2 nat_3)
                       3))

    '((sum (S (S Z)) (S (S Z)) (S (S (S (S Z)))))

      (sum (S (S Z)) Z (S (S Z)))

      (sum (S (S (S Z))) Z (S (S (S Z))))

      (sum Z Z Z)

      (sum (S (S (S (S Z)))) Z (S (S (S (S Z)))))

      (sum Z Z Z)

      (sum (S (S (S Z))) (S (S Z)) (S (S (S (S (S Z))))))

      (sum (S Z) (S (S Z)) (S (S (S Z))))

      (sum (S (S (S Z))) (S Z) (S (S (S (S Z)))))

      (sum (S (S Z)) (S (S Z)) (S (S (S (S Z))))))

  • from-metafunction: The first form randomly picks a term that satisfies the given invocation of the metafunction, using techniques similar to how the from-judgment-form case works. The second form uses a more naive approach; it simply generates terms that match the patterns of the cases of the metafunction; it does not consider the results of the metafunctions, nor does it consider patterns from earlier cases when generating terms based on a particular case. The third case is like the second, except it returns a function that accepts the size and keywords arguments that may be more efficient if multiple random terms are generated.

    Examples:
    > (define-language L
       (n number))
    > (define-metafunction L
        [(F one-clause n) ()]
        [(F another-clause n) ()])
    > (for/list ([i (in-range 10)])
        (generate-term #:source F 5))

    '((another-clause 2)

      (another-clause 4)

      (one-clause 1)

      (another-clause 1)

      (one-clause 0)

      (another-clause 0)

      (another-clause 0)

      (another-clause 3)

      (another-clause 1)

      (another-clause 10))

  • from-reduction-relation: In the first case, generate-term randomly picks a rule from the reduction relation and tries to pick a term that satisfies its domain pattern, returning that. The second case returns a function that accepts the size and keyword arguments that may be more efficient if multiple random terms are generated.

    Examples:
    > (define-language L
       (n number))
    > (for/list ([i (in-range 10)])
        (generate-term
         #:source
         (reduction-relation
          L
          (--> (one-clause n) ())
          (--> (another-clause n) ()))
         5))

    '((one-clause 2)

      (one-clause 0)

      (another-clause 2)

      (another-clause 0)

      (another-clause 4)

      (another-clause 2)

      (one-clause 5)

      (one-clause 0)

      (one-clause 0)

      (another-clause 0))

The argument size-expr bounds the height of the generated term (measured as the height of its parse tree).

The optional keyword argument attempt-num-expr (default 1) provides coarse grained control over the random decisions made during generation; increasing attempt-num-expr tends to increase the complexity of the result. For example, the absolute values of numbers chosen for integer patterns increase with attempt-num-expr.

The random generation process does not actively consider the constraints imposed by side-condition or _!_ patterns; instead, it uses a “guess and check” strategy in which it freely generates candidate terms then tests whether they happen to satisfy the constraints, repeating as necessary. The optional keyword argument retries-expr (default 100) bounds the number of times that generate-term retries the generation of any pattern. If generate-term is unable to produce a satisfying term after retries-expr attempts, it raises an exception recognized by exn:fail:redex:generation-failure?.

syntax

(redex-enum language pattern)

Constructs an enumeration that produces terms that match the given pattern, or #f if it cannot build an enumeration (which happens if the given pattern contains side-conditions).

It constructs a two-way enumeration only in some cases. The pattern must be unambiguous and there are other technical shortcomings of the implementation as well that cause the result to be a one-way enumeration in some situations.

Examples:
> (define-language L
    (e ::= (e e) x (λ (x) e))
    (x ::= variable-not-otherwise-mentioned))
> (from-nat (redex-enum L e) 3886654839907963757723234276487685940)

'(λ (f) (f (f (f x))))

> (from-nat (redex-enum L e) 3886654839907963757723234276487685942)

'(c (f (f (f x))))

> (from-nat (redex-enum L e) 3886654839907963757723234276487685945)

'(((a a) a) (f (f (f x))))

> (to-nat (redex-enum L e)
          (term (λ (f) ((λ (x) (f (x x)))
                        (λ (x) (f (x x)))))))

68069248527054969607740967414758416542534392945933384867539062268798775005

syntax

(redex-index language pattern term)

Computes the index for an occurrence of the given term in the enumerator corresponding to the given pattern or returns #f if there is no enumerator.

This is useful when the pattern is ambiguous as you might still learn of an index that corresponds to the term even though the enumeration that redex-enum produces is a one-way enumeration.

Examples:
> (define-language L
    (e ::= (e e) x (λ (x) e))
    (x ::= variable-not-otherwise-mentioned))
> (redex-index L e
               (term (λ (f) ((λ (x) (f (x x)))
                             (λ (x) (f (x x)))))))

68069248527054969607740967414758416542534392945933384867539062268798775005

> (define-language L
    ; e is an ambiguous non-terminal
    ; because there are multiple ways to
    ; parse (cons (λ (x) x) (λ (x) x))
    (e ::= (e e) x (cons e e) v)
    (v ::= (cons v v) (λ (x) e))
    (x ::= variable-not-otherwise-mentioned))
> (redex-index L e
               (term ((λ (x) (x x))
                      (λ (x) (x x)))))

366642754084173391502

syntax

(redex-check template property-expr kw-arg ...)

 
template = language pattern
  | language pattern #:ad-hoc
  | language pattern #:in-order
  | language pattern #:uniform-at-random p-value
  | language pattern #:enum bound
  | 
language #:satisfying
(judgment-form-id pattern ...)
  | 
language #:satisfying
(metafunction-id pattern ...) = pattern
     
kw-arg = #:attempts attempts-expr
  | #:source metafunction
  | #:source relation-expr
  | #:retries retries-expr
  | #:print? print?-expr
  | #:attempt-size attempt-size-expr
  | #:prepare prepare-expr
  | #:keep-going? keep-going?-expr
 
  property-expr : any/c
  attempts-expr : natural-number/c
  relation-expr : reduction-relation?
  retries-expr : natural-number/c
  print?-expr : any/c
  attempt-size-expr : (-> natural-number/c natural-number/c)
  prepare-expr : (-> any/c any/c)
Searches for a counterexample to property-expr, interpreted as a predicate universally quantified over the pattern variables bound by the pattern(s) in template. redex-check constructs and tests a candidate counterexample by choosing a random term t based on template and then evaluating property-expr using the match-bindings produced by matching t against pattern. The form of template controls how t is generated:
  • language pattern: In this case, redex-check uses an ad hoc strategy for generating pattern. For the first 10 seconds, it uses in-order enumeration to pick terms. After that, it alternates back and forth between in-order enumeration and the ad hoc random generator. After the 10 minute mark, it switches over to using just the ad hoc random generator.

  • language pattern #:ad-hoc: In this case, redex-check uses an ad hoc random generator to generate terms that match pattern.

  • language pattern #:in-order: In this case, redex-check uses an enumeration of pattern, checking each t one at a time

  • language pattern #:uniform-at-random p-value: that to index into an enumeration of pattern. If the enumeration is finite, redex-check picks a natural number uniformly at random; if it isn’t, redex-check uses a geometric distribution with p-value as its probability of zero to pick the number of bits in the index and then picks a number uniformly at random with that number of bits.

  • language pattern #:enum bound: This is similar to #:uniform-at-random, except that Redex always picks a random natural number less than bound to index into the enumeration

  • language #:satisfying (judgment-form-id pattern ...): Generates terms that match pattern and satisfy the judgment form.

  • language #:satisfying (metafunction-id pattern ...) = pattern: Generates terms matching the two patterns, such that if the first is the argument to the metafunction, the second will be the result.

redex-check generates at most attempts-expr (default (default-check-attempts)) random terms in its search. The size and complexity of these terms tend to increase with each failed attempt. The #:attempt-size keyword determines the rate at which terms grow by supplying a function that bounds term size based on the number of failed attempts (see generate-term’s size-expr argument). By default, the bound grows according to the default-attempt-size function.

When print?-expr produces any non-#f value (the default), redex-check prints the test outcome on current-output-port. When print?-expr produces #f, redex-check prints nothing, instead
  • returning a counterexample structure when the test reveals a counterexample,

  • returning #t when all tests pass, or

  • raising a exn:fail:redex:test when checking the property raises an exception.

The optional #:prepare keyword supplies a function that transforms each generated example before redex-check checks property-expr. This keyword may be useful when property-expr takes the form of a conditional, and a term chosen freely from the grammar is unlikely to satisfy the conditional’s hypothesis. In some such cases, the prepare keyword can be used to increase the probability that an example satisfies the hypothesis.

The #:retries keyword behaves identically as in generate-term, controlling the number of times the generation of any pattern will be reattempted. It can’t be used together with #:satisfying.

If keep-going?-expr produces any non-#f value, redex-check will stop only when it hits the limit on the number of attempts showing all of the errors it finds. This argument is allowed only when print?-expr is not #f.

When passed a metafunction or reduction relation via the optional #:source argument, redex-check distributes its attempts across the left-hand sides of that metafunction/relation by using those patterns, rather than pattern, as the basis of its generation. If any left-hand side generates a term that does not match pattern, then the test input is discarded. #:source cannot be used with #:satisfying. See also check-reduction-relation and check-metafunction.

Examples:
> (define-language empty-lang)
> (random-seed 0)
> (redex-check
   empty-lang
   ((number_1 ...)
    (number_2 ...))
   (equal? (reverse (append (term (number_1 ...))
                            (term (number_2 ...))))
           (append (reverse (term (number_1 ...)))
                   (reverse (term (number_2 ...))))))

redex-check: counterexample found after 11 attempts:

((+inf.0) (0))

> (redex-check
   empty-lang
   ((number_1 ...)
    (number_2 ...))
   (equal? (reverse (append (term (number_1 ...))
                            (term (number_2 ...))))
           (append (reverse (term (number_2 ...)))
                   (reverse (term (number_1 ...)))))
   #:attempts 200)

redex-check: no counterexamples in 200 attempts

> (let ([R (reduction-relation
            empty-lang
            (--> (Σ) 0)
            (--> (Σ number) number)
            (--> (Σ number_1 number_2 number_3 ...)
                 (Σ ,(+ (term number_1) (term number_2))
                    number_3 ...)))])
    (redex-check
     empty-lang
     (Σ number ...)
     (printf "~s\n" (term (number ...)))
     #:attempts 3
     #:source R))

()

(0)

(0 2)

redex-check: no counterexamples in 3 attempts (tried 1 attempt with each clause)

> (redex-check
   empty-lang
   number
   (begin
     (printf "checking ~s\n" (term number))
     (positive? (term number)))
   #:prepare (λ (n)
               (printf "preparing ~s; " n)
               (add1 (abs (real-part n))))
   #:attempts 3)

preparing +inf.0; checking +inf.0

preparing 0; checking 1

preparing +inf.0+inf.0i; checking +inf.0

redex-check: no counterexamples in 3 attempts

> (define-language L
    (nat ::= Z (S nat)))
> (define-judgment-form L
    #:mode (sum I I O)
    [---------------
     (sum Z nat nat)]
    [(sum nat_1 nat_2 nat_3)
     -------------------------------
     (sum (S nat_1) nat_2 (S nat_3))])
> (redex-check L
               #:satisfying
               (sum nat_1 nat_2 nat_3)
               (equal? (judgment-holds
                        (sum nat_1 nat_2 nat_4) nat_4)
                       (term (nat_3)))
               #:attempts 100)

redex-check: no counterexamples in 100 attempts

> (redex-check L
               #:satisfying
               (sum nat_1 nat_2 nat_3)
               (equal? (term nat_1) (term nat_2)))

redex-check: counterexample found after 1 attempt:

(sum (S (S Z)) (S Z) (S (S (S Z))))

Changed in version 1.10 of package redex-lib: Instead of raising an error, redex-check now discards test cases that don’t match the given pattern when using #:source.

parameter

(depth-dependent-order?)  (or/c boolean? 'random)

(depth-dependent-order? depth-dependent)  void?
  depth-dependent : (or/c boolean? 'random)
 = 'random
Toggles whether or not Redex will dynamically adjust the chance that more recursive clauses of judgment forms or metafunctions are chosen earlier when attempting to generate terms with forms that use #:satisfying. If it is #t, Redex favors more recursive clauses at lower depths and less recursive clauses at depths closer to the limit, in an attempt to generate larger terms. When it is #f, all clause orderings have equal probability above the bound. By default, it is 'random, which causes Redex to choose between the two above alternatives with equal probability.

syntax

(redex-generator language-id satisfying size-expr)

 
satisfying = (judgment-form-id pattern ...)
  | (metafunction-id pattern ...) = pattern
 
  size-expr : natural-number/c
WARNING: redex-generator is a new, experimental form, and its API may change.

Returns a thunk that, each time it is called, either generates a random s-expression based on satisfying or fails to (and returns #f). The terms returned by a particular thunk are guaranteed to be distinct.

Examples:
> (define-language L
    (nat ::= Z (S nat)))
> (define-judgment-form L
    #:mode (sum I I O)
    [---------------
     (sum Z nat nat)]
    [(sum nat_1 nat_2 nat_3)
     -------------------------------
     (sum (S nat_1) nat_2 (S nat_3))])
> (define gen-sum (redex-generator L (sum nat_1 nat_2 nat_3) 3))
> (for/list ([_ (in-range 5)])
    (gen-sum))

'((sum (S (S (S Z))) (S (S Z)) (S (S (S (S (S Z))))))

  (sum (S (S (S (S Z)))) (S Z) (S (S (S (S (S Z))))))

  (sum (S (S (S (S (S Z))))) Z (S (S (S (S (S Z))))))

  (sum (S (S (S (S (S (S Z)))))) (S (S Z)) (S (S (S (S (S (S (S (S Z)))))))))

  (sum (S (S (S (S (S (S (S Z))))))) (S Z) (S (S (S (S (S (S (S (S Z))))))))))

struct

(struct counterexample (term)
    #:extra-constructor-name make-counterexample
    #:transparent)
  term : any/c
Produced by redex-check, check-reduction-relation, and check-metafunction when testing falsifies a property.

struct

(struct exn:fail:redex:test exn:fail:redex (source term)
    #:extra-constructor-name make-exn:fail:redex:test)
  source : exn:fail?
  term : any/c
Raised by redex-check, check-reduction-relation, and check-metafunction when testing a property raises an exception. The exn:fail:redex:test-source component contains the exception raised by the property, and the exn:fail:redex:test-term component contains the term that induced the exception.

syntax

(check-reduction-relation relation property kw-args ...)

 
kw-arg = #:attempts attempts-expr
  | #:retries retries-expr
  | #:print? print?-expr
  | #:attempt-size attempt-size-expr
  | #:prepare prepare-expr
 
  property : (-> any/c any/c)
  attempts-expr : natural-number/c
  retries-expr : natural-number/c
  print?-expr : any/c
  attempt-size-expr : (-> natural-number/c natural-number/c)
  prepare-expr : (-> any/c any/c)
Tests relation as follows: for each case of relation, check-reduction-relation generates attempts random terms that match that case’s left-hand side and applies property to each random term.

Only the primary pattern of each case’s left-hand side is considered. If there are where clauses or side-conditions (or anything else from the red-extras portion of the grammar), they are ignored.

This form provides a more convenient notation for
(redex-check L any (property (term any))
             #:attempts (* n attempts)
             #:source relation)
when relation is a relation on L with n rules.

syntax

(check-metafunction metafunction property kw-args ...)

 
kw-arg = #:attempts attempts-expr
  | #:retries retries-expr
  | #:print? print?-expr
  | #:attempt-size attempt-size-expr
  | #:prepare prepare-expr
 
  property : (-> (listof any/c) any/c)
  attempts-expr : natural-number/c
  retries-expr : natural-number/c
  print?-expr : any/c
  attempt-size-expr : (-> natural-number/c natural-number/c)
  prepare-expr : (-> (listof any/c) (listof any/c))
Like check-reduction-relation but for metafunctions. check-metafunction calls property with lists containing arguments to the metafunction. Similarly, prepare-expr produces and consumes argument lists.

Only the primary pattern of each case’s left-hand side is considered. If there are where clauses or side-conditions (or anything else from the metafunction-extras portion of the grammar), they are ignored.

Examples:
> (define-language empty-lang)
> (define-metafunction empty-lang
    Σ : number ... -> number
    [(Σ) 0]
    [(Σ number) number]
    [(Σ number_1 number_2) ,(+ (term number_1) (term number_2))]
    [(Σ number_1 number_2 ...) (Σ number_1 (Σ number_2 ...))])
> (check-metafunction Σ (λ (args)
                          (printf "trying ~s\n" args)
                          (equal? (apply + args)
                                  (term (Σ ,@args))))
                      #:attempts 2)

trying ()

trying ()

trying (0)

trying (0)

trying (2 1)

trying (0 1)

trying (0)

trying (1)

check-metafunction: no counterexamples in 8 attempts (tried 2 attempts with each clause)

The default value of the #:attempt-size argument to redex-check and the other randomized testing forms, this procedure computes an upper bound on the size of the next test case from the number of previously attempted tests n. Currently, this procedure computes the base 5 logarithm, but that behavior may change in future versions.

parameter

(default-check-attempts)  natural-number/c

(default-check-attempts attempts)  void?
  attempts : natural-number/c
Determines the default value for redex-check’s optional #:attempts argument. By default, attempts is 1,000.

generate-term and the randomized testing forms (e.g., redex-check) use the parameter generator to construct random terms. The parameter’s initial value is (current-pseudo-random-generator).

Recognizes the exceptions raised by generate-term, redex-check, etc. when those forms are unable to produce a term matching some pattern.

Debugging PLT Redex Programs

It is easy to write grammars and reduction rules that are subtly wrong. Typically such mistakes result in examples that get stuck when viewed in a traces window.

The best way to debug such programs is to find an expression that looks like it should reduce, but doesn’t, then try to find out which pattern is failing to match. To do so, use the redex-match form.

In particular, first check if the term in question matches the your system’s main non-terminal (typically the expression or program non-terminal). If it does not match, simplify the term piece by piece to determine whether the problem is in the term or the grammar.

If the term does match your system’s main non-terminal, determine by inspection which reduction rules should apply. For each such rule, repeat the above term-pattern debugging procedure, this time using the rule’s left-hand side pattern instead of the system’s main non-terminal. In addition to simplifying the term, also consider simplifying the pattern.

If the term matches the left-hand side, but the rule does not apply, then one of the rule’s side-condition or where clauses is not satisfied. Using the bindings reported by redex-match, check each side-condition expression and each where pattern-match to discover which clause is preventing the rule’s application.

4.7 GUI

 (require redex/gui) package: redex-gui-lib

This section describes the GUI tools that Redex provides for exploring reduction sequences.

procedure

(traces reductions 
  expr 
  [#:multiple? multiple? 
  #:reduce reduce 
  #:pred pred 
  #:pp pp 
  #:colors colors 
  #:racket-colors? racket-colors? 
  #:scheme-colors? scheme-colors? 
  #:filter term-filter 
  #:x-spacing x-spacing 
  #:y-spacing y-spacing 
  #:layout layout 
  #:edge-labels? edge-labels? 
  #:edge-label-font edge-label-font 
  #:graph-pasteboard-mixin graph-pasteboard-mixin]) 
  void?
  reductions : (or/c reduction-relation? IO-judgment-form?)
  expr : (or/c any/c (listof any/c))
  multiple? : boolean? = #f
  reduce : 
(-> reduction-relation? any/c
    (listof (list/c (union false/c string?) any/c)))
   = apply-reduction-relation/tag-with-names
  pred : 
(or/c (-> sexp any)
      (-> sexp term-node? any))
 = (λ (x) #t)
  pp : 
(or/c (any -> string)
      (any output-port number (is-a?/c text%) -> void))
   = default-pretty-printer
  colors : 
(listof
 (cons/c string?
         (and/c (listof (or/c string? (is-a?/c color%)))
                (λ (x) (<= 0 (length x) 6)))))
   = '()
  racket-colors? : boolean? = #t
  scheme-colors? : boolean? = racket-colors?
  term-filter : (-> any/c (or/c #f string?) any/c)
   = (λ (x y) #t)
  x-spacing : real? = 15
  y-spacing : real? = 15
  layout : (-> (listof term-node?) void?) = void
  edge-labels? : boolean? = #t
  edge-label-font : (or/c #f (is-a?/c font%)) = #f
  graph-pasteboard-mixin : (make-mixin-contract graph-pasteboard<%>)
   =