4 Reduction Relations

The optional domain clause provides a contract for the relation, in the form of a pattern that defines the relation’s domain and codomain.

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 ((lambda (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. The bindings are the same as bindings in a term-let expression. A where/hidden clause is the same as a where clause, but the clause is not rendered when typesetting via redex/pict.

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.

For example, this expression

(reduction-relation

lc-num-lang

(==> ((lambda (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 lambda 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.

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.

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).

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)